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Control and optimization of complex networked systems: wireless communication and power grids

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Control and optimization of complex networked systems: wireless communication and power grids

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Content ControlandOptimizationofComplex NetworkedSystems WirelessCommunication and Power Grids Reza Banirazi MingHsiehDepartmentofElectricalEngineering UniversityofSouthernCalifornia ADissertationPresentedtothe FACULTYOF THEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA inPartialFulfillmentoftheRequirementsfortheDegree DoctorofPhilosophy (ELECTRICALENGINEERING) ViterbiSchoolofEngineering December2018 Thisthesisisdedicatedto myparentsfortheirunconditionalloveandsupport. Acknowledgements Ithasbeena greathonor andprivilegetoworkwithmyadvisorProf. EdmondJonckheere, who Iamforeverindebtedtoforshapingmyacademiclifeandbeyond. Iwouldliketoexpressmy deepest gratitude to him for his guidance, patience, support and encouragement. He taught me howtoconducthigh-impactinterdisciplinaryresearchonabroadrangeofprojects,andmore importantly, howto stay enthusiastic andmotivated during the toughtimes of research. Thank youProf. Jonckheere forhaving meso closeto yourselfthroughout theseyears andgiving me theopportunitytolearnyourteachingandresearchmethods. IamgratefultoProf. BhaskarKrishnamachariforintroducingmetothechallengingarea of wireless networking, and for mentoring me how to connect theory to practice and present results in a clear and concise manner. He has been a constant source of critical discussions and halfofthisresearchhasbeendevelopedunderthegreatsupervisionofhim. My special thanks go to Prof. Francis Bonahon for helping me with difficult concepts in hyperbolic geometry. At the time he was chair of USC Department of Mathematics and still foundtimetotalktome. Iwouldliketothankmyqualifyingcommittee,Prof. MichaelSafonov,UrbashiMitraand RahulJainfortheirhelpfulguidanceandcomments. IalsothankProf. MichaelNeelyforhis constructivefeedbackanddiscussionsinreferencetowirelessnetworkoptimization. Abstract Complex networked systems appear in almost every aspect of science and technology. Most social, biological, informational and physical networks are large assemblies of nonlinear dy- namical systems interacting via non-trivial topologies. Many of these networks are complex in different ways including that they are large in scale and so hard to visualize; they possess non-trivialstructureofconnectionsbetweentheirelements,whichareneitherpurelyregular norpurelyrandom;andtheyoperateundermodelinguncertaintiesandstochasticdisturbances. Managing complex networks is challenging due to the ambiguity of what is really essential for overallperformance,reliabilityandsecurity. Itisalsounclearwhatnewphenomenawillemerge asa resultof theinterplaybetween topologyand traffic. Theobjectiveof thisdissertation is to tackle some of these challenges, where the main focus is motivated by applied problems in two importantfieldsofwirelessnetworksandpowergrids. Inthefieldofwirelessnetworks,wedevelop,analyzeandevaluatenewmulti-classdynamic resource allocation and routing algorithms, assuming stochastic arrivals, time-varying topology andinter-channelinterference. Besidesthroughput optimality, ourapproach allows forPareto optimality relative to average quadratic routing cost and average network delay. In a somewhat simplistic metaphorical view of the packet network where queue lengths are heat quantities or electrical voltages and packets are calories or electrons, the flow of packets under our control policies takes the form of heat flow or electrical currents towards the sink or ground. Thisdevelopsanewparadigm,whichmightbecalled“WirelessNetworkThermodynamics,” by building a rigorous connection between wireless networking and important domains in mathematical physics. In particular, it opens a way to take advantage of powerful tools from circuit theory, such as effective resistance and graph Laplacian, and those from differential geometry,such as heatcalculus and curvature,in theanalysis and designof wireless networks, ormoregenerallystochasticoperationalprocessingnetworks. We bring to bear several novel methods to study the problems in this research. The first methodisthatofstochasticoptimization,wheretheclassicalsymmetricpositivedefiniteLya- viii punov function fails on the Pareto-optimized policy, requiring unorthodox Lyapunov techniques to prove routing stability and optimality. The second method of fluid limit theory approximates the evolution of stochastic interference flow of packets with a deterministic continuous-time processusingsomescalingandlimitingcriteria. Thethirdmethoddescribesthefluidapproxima- tionofoptimizedroutingpolicyasthesuperpositionofasmanynon-interferencediffusionflows as there are packet classes. This brings nonlinear Laplacian methods, where an entirely new paradigm of multiclass conduction on capacitated directed graphs is proposed. This, combined with the fourth method of heat calculus, proves optimality relative to routing cost as a corollary of the Dirichlet principle. The fifth method of multi-objective optimization is proposed to prove Pareto-optimalitywithrespecttonetworkdelayandroutingcost. Finally,thesixthmethodis that of curvature as a mean to evaluate network stability region with respect to the network topology. ThislastmethodcallsforthefutureresearchchallengesofvalidatingtheOllivier-Ricci curvature on Finsler manifolds, as a prelude to its validation on directed graphs, and ultimately itsvalidationasaheatdiffusionperformancepredictor. The broader impact of this research includes its potential application to other fields than wirelessnetworksandoperationalprocessingnetworks. Forinstance,insomefuturisticsmart grid applications where massive energystoragetechnology willbe available, onecan certainly envisionanalternativeofourproposedcontrolpoliciesforenergyrouting. Stillrelatedtothe powergrid,theOllivier-Riccicurvaturethatcanbeusedheretofleshoutnetworkstabilityis theperfectmeasureofthecostoftransportingpowerfromgenerationstationstodistribution centers. Stochasticresourceallocation problemsinmanufacturing andtransportationsystems fallwithinthescopeofthisresearch aswell. In the field of power grids, we develop a smart grid concept where the fluctuation of power generationfromdistributedrenewablesourcesismanaged,atalarge-scalelevel,toevenlytransfer the power over the network without overloading some critical transmission lines or excessively relyingonsomecriticalstations. Themethodisthatofcurvatureinthelarge-scalegeometry sense,whichtradeslocaldetailsforabetterunderstandingofglobalproperties. Adoptingthe conceptofeffectiveresistance,weproposeanew electricalengineeringdefinitionofnegative curvature,remotefromthemathematicalrepertoire,basedontheasymptoticbehaviorofthe ratio of the path resistance to the effective resistance. This brings coarse Riemannian geometry in an arena it has not yet pervaded—power networks. From a deeper conceptual standpoint, however, it naturally leads to transport problems where, contrary to the classical operational research problems, the commodity is characterized by two conjugate Hamiltonian variables and itispropagatedalongasmanypaths aspossiblewithnoprescription. Tableofcontents Listoffigures xv 1 Introduction 1 1.1 CombinatorialGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ComplexNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 StochasticResourceOptimization . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 CommunicationNetworksversusPowerGrids . . . . . . . . . . . . . . . . . 4 1.5 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I WirelessNetworks 9 2 SmoothversusCombinatorialGeometry 11 2.1 SmoothGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Topologicalmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Tangentandcotangentspaces . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Metrictensorandinnerproduct . . . . . . . . . . . . . . . . . . . . 14 2.1.4 Pathsandgeodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.5 Riemannianmeasure . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.6 Gradient,divergenceandLaplaceoperators . . . . . . . . . . . . . . 16 2.1.7 Weightedmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 CombinatorialGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Cellcomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Chainsandcochains . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 Metrictensorandinnerproductoncomplex . . . . . . . . . . . . . . 20 2.2.4 Combinatorialexteriorderivative . . . . . . . . . . . . . . . . . . . . 21 x Tableofcontents 2.2.5 Homologyandcohomology . . . . . . . . . . . . . . . . . . . . . . 23 2.2.6 Incidencematrixanddualcomplex . . . . . . . . . . . . . . . . . . 24 2.2.7 CombinatorialHodgeduality . . . . . . . . . . . . . . . . . . . . . 25 2.2.8 Combinatorialcodifferentialoperator . . . . . . . . . . . . . . . . . 26 2.2.9 CombinatorialLaplace-deRhamoperator . . . . . . . . . . . . . . . 26 2.2.10 Weightedcomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Graphas1-complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Resistivenetworkinterpretation . . . . . . . . . . . . . . . . . . . . 29 3 Heat-Diffusion: ParetoOptimalDynamicRouting 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 Relatedworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.1 Problemstatement . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.2 UniclassBack-Pressure(BP)policy . . . . . . . . . . . . . . . . . . . 37 3.2.3 UniclassV-parameterBPpolicy . . . . . . . . . . . . . . . . . . . . 38 3.3 UniclassHeat-Diffusion(HD)Policy . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 UniclassParetooptimalHDalgorithm . . . . . . . . . . . . . . . . . 39 3.3.2 HighlightsofParetooptimalHDdesign . . . . . . . . . . . . . . . . 40 3.3.3 Illustrativeexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 KeyPropertyofParetoOptimality . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 HDThroughputOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.1 Characteristicofnetworkcapacityregion . . . . . . . . . . . . . . . 49 3.5.2 HDthroughputoptimalityforallβ . . . . . . . . . . . . . . . . . . . 50 3.6 HDMinimumDelayatβ =0 . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7 ClassicalvsCombinatorialHeatProcess . . . . . . . . . . . . . . . . . . . . 63 3.7.1 Heatequationsonmanifolds . . . . . . . . . . . . . . . . . . . . . . 63 3.7.2 Heatequationsonundirectedgraphs . . . . . . . . . . . . . . . . . . 64 3.7.3 Heatequationsondirectedgraphs . . . . . . . . . . . . . . . . . . . 65 3.8 WirelessNetworkThermodynamics . . . . . . . . . . . . . . . . . . . . . . 65 3.8.1 ParetooptimalHDfluidlimit . . . . . . . . . . . . . . . . . . . . . 66 Tableofcontents xi 3.8.2 Thermodynamic-likepacketrouting . . . . . . . . . . . . . . . . . . 69 3.9 HDMinimumRoutingCostatβ =1 . . . . . . . . . . . . . . . . . . . . . . 73 3.9.1 ClassicalDirichletprinciple . . . . . . . . . . . . . . . . . . . . . . 73 3.9.2 CombinatorialDirichletprinciple . . . . . . . . . . . . . . . . . . . 74 3.9.3 NonlinearDirichletprinciple . . . . . . . . . . . . . . . . . . . . . . 74 3.9.4 Quadraticroutingcostminimization . . . . . . . . . . . . . . . . . . 76 3.10 ParetoOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.10.1 StrongParetooptimalityfornonuniformlinkcosts . . . . . . . . . . 79 3.10.2 WeakParetooptimalityforunitlinkcosts . . . . . . . . . . . . . . . 80 3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 MulticlassMinimumDelayRouting 83 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.1 Relatedworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.1 Systemdescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.2 Networkcapacityregion . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2.3 MulticlassBack-Pressure(BP)policy . . . . . . . . . . . . . . . . . . 91 4.3 MinimumDelayPolicy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.1 MulticlassminimumdelaysubjecttoSCLT . . . . . . . . . . . . . . 92 4.3.2 MulticlassminimumdelaywithMCLT . . . . . . . . . . . . . . . . 94 4.4 DynamicCharacteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5 ThroughputAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.6 RoutingDelayAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.7 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.7.1 Uniclasssensornetwork . . . . . . . . . . . . . . . . . . . . . . . . 108 4.7.2 Multiclassmeshnetwork . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 MulticlassMinimumCostRouting 115 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1.1 DiscussionontheDirichletroutingcost . . . . . . . . . . . . . . . . 116 5.1.2 Relatedworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 xii Tableofcontents 5.1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2.1 Stabilityandthroughputoptimality . . . . . . . . . . . . . . . . . . 120 5.2.2 Inter-channelinterference . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2.3 Time-varyingtopology . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2.4 Statemodelofmulticlassnetworks . . . . . . . . . . . . . . . . . . 122 5.2.5 MulticlassV-parameterBPpolicy . . . . . . . . . . . . . . . . . . . 124 5.3 NonlinearConduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3.1 Conductiononmanifolds . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3.2 Conductiononresistivenetworks . . . . . . . . . . . . . . . . . . . 126 5.3.3 Conductiononnonlinearresistivenetworks . . . . . . . . . . . . . . 128 5.4 Multi-chargeNonlinearConduction . . . . . . . . . . . . . . . . . . . . . . 133 5.5 Dirichlet-BasedRoutingPolicy . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6 RoutingCostAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.6.1 Dirichlet-basedtimeslotbehavior . . . . . . . . . . . . . . . . . . . 143 5.6.2 Dirichlet-basedfluidbehavior . . . . . . . . . . . . . . . . . . . . . 145 5.7 ThroughputAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.8 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.9 ConclusionandFutureResearch . . . . . . . . . . . . . . . . . . . . . . . . 156 II PowerGrids 159 6 CombinatorialGraphCurvature 161 6.1 CurvatureFlowonRiemannianSurfaces . . . . . . . . . . . . . . . . . . . . . 161 6.1.1 RiemanniansurfaceandGauss-Bonnettheorem . . . . . . . . . . . . 162 6.1.2 Conformalequivalence . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1.3 SmoothsurfaceRicciflow . . . . . . . . . . . . . . . . . . . . . . . 164 6.2 GraphCurvatureintheLocal . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.2.1 PiecewiseLinearGeometry . . . . . . . . . . . . . . . . . . . . . . 166 6.2.2 Localcurvatureforplanargraphs . . . . . . . . . . . . . . . . . . . . 167 6.2.3 Localcurvaturefornon-planargraphs . . . . . . . . . . . . . . . . . 168 6.2.4 Fromgeometry(local)totopology(global) . . . . . . . . . . . . . . 169 Tableofcontents xiii 6.2.5 Reliabilityoflocalcurvature . . . . . . . . . . . . . . . . . . . . . . 170 6.3 CurvatureFlowonPlanarGraphs . . . . . . . . . . . . . . . . . . . . . . . . 171 6.3.1 Circlepacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.3.2 CombinatorialRicciflow . . . . . . . . . . . . . . . . . . . . . . . 172 6.3.3 Conformalmap computation . . . . . . . . . . . . . . . . . . . . . . 174 6.4 GraphCurvatureintheLarge . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.5 CurvatureversusCongestion . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7 NegativelyCurvedPowerGrids 181 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2 DynamicModelofPowerSystem . . . . . . . . . . . . . . . . . . . . . . . 183 7.2.1 Multi-time-scale modeling . . . . . . . . . . . . . . . . . . . . . . . 183 7.2.2 Time-scaledecompositionofpowersystems . . . . . . . . . . . . . . 185 7.2.3 Powerflowequations . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.3 VirtualNetworksofPowerGrid . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3.1 Intuitiveidea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.3.2 Virtualnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.3.3 Electricalversusgeodesicdistance . . . . . . . . . . . . . . . . . . . . 191 7.4 GeometryofPowerTransmission . . . . . . . . . . . . . . . . . . . . . . . . 192 7.4.1 Hyperbolicresistivenetwork . . . . . . . . . . . . . . . . . . . . . . 192 7.4.2 Gridcurvatureversuslineoverload . . . . . . . . . . . . . . . . . . 194 7.4.3 Effectoffluctuationofrenewables . . . . . . . . . . . . . . . . . . . . 197 7.5 EvaluationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.6 SmartPowerSchedulingandRouting . . . . . . . . . . . . . . . . . . . . . 200 7.6.1 Ricciflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.6.2 Star-deltatransformations . . . . . . . . . . . . . . . . . . . . . . . 202 7.6.3 Triangulationcondition . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.6.4 ConstrainedRicciflow . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.6.5 Localversusglobal . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.8 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.8.1 Developmentofarigorousmathematicaltheory . . . . . . . . . . . . 205 7.8.2 Topologyimpactonstateobservabilityofpowergrid . . . . . . . . . 205 7.8.3 Topologyimpactonelectricalsecurityofpowergrid . . . . . . . . . 206 xiv Tableofcontents 7.8.4 Powerflowcontrolversusvoltagestability . . . . . . . . . . . . . . . . 207 8 Multi-AgentCooperativeVoltageControl 209 8.1 LargeScaleVoltageStability . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.3.1 AgentArchitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.3.2 EmergencyModeOperation . . . . . . . . . . . . . . . . . . . . . . 212 8.3.3 NormalModeOperation . . . . . . . . . . . . . . . . . . . . . . . . 212 References 215 Listoffigures 2.1 Primal anddual cells, alongwiththecombinatorialmappingoperatorsona3- complex. There exists a dual 3-cochain (polyhedron) for each primal 0-cochain (vertex), a dual 2-cochain (polygon) for each primal 1-cochain (edge), a dual 1-cochain for each primal 2-cochain, and a dual 0-cochain for each primal 3- cochain. The coboundary operatord and the codifferential operator δ map betweenk and(k+1) cochainsonthesamecomplexofeitherprimalordual, while the Hodge star operator ⋆ and its inverse ⋆ − 1 swap cochains between primalanddualcomplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 GraphicaldescriptionofHDParetooptimalitywithrespecttoaveragequeue congestionandtheDirichletroutingcost,comparedwiththeperformanceof V-parameterBP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Two-queuedownlink: PerformanceofHDwithβ =0versusoriginalBP.While foralladmissiblelinkcapacitiestotalqueueisminimizedunderHD,itgrows linearlyinµ 2 underBP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Lossy link network: Performance of Pareto optimal HD versus V-parameter BP. While total queued packetsisstabilizedat1underHDforanyβ > 0,itgrows linearlyinV underBP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Power minimization: Timeslot total queue backlog in HD withβ = 0 versus original BP, showing the minimization of average queue congestion by HD. Noticeable is also the little steady-state oscillations in total queue under HD contrarytoitslargevariationsunderBP. . . . . . . . . . . . . . . . . . . . . 44 3.5 Power minimization: Timeslot total power consumption, which is highly corre- latedwiththeDirichletroutingcost,inParetooptimalHDwithβ =1versus V-parameterBPwithV =10. . . . . . . . . . . . . . . . . . . . . . . . . . 45 xvi Listoffigures 3.6 Power minimization: Queue congestion versus power consumption in Pareto optimalHDagainstβ andinV-parameterBPagainstV,withthedashedlines representinginterpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 GraphicaldescriptionofweakParetoboundarywithrespecttoaveragequeue congestion and the Dirichlet routing cost when all link cost factors converge to one,contrastingtheperformanceofParetooptimalHDwithV-parameterBP. 80 4.1 Geometricaldescriptionofproblem (4.14)foratwo-classcasewithq (1) ij (n)+ q (2) ij (n) > µ ij (n), showing that the integer problem can have more than one optimalsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Simulationtestbenchwith50nodesrandomlydistributedonasurface. Links are placed between every two offset nodes with a proximity distance less than a thresholdandextralinks withminimumlengthareaddedtomake thenetwork connected,whichresultin128two-waywirelesslinks. Thenodewitharing indicatesthesinkforthecaseofuniclasssensornetwork. . . . . . . . . . . . 108 4.3 Timeslotevolutionoftotalnumberofpacketsintheuniclasssensornetwork, displaying the performance of minimum delay policy versus BP policy for two arrivalratesofλ =1(left)andλ =10(right)ateachnode. Thelowerpanelsare zoomed in the 0-2000 timeslot interval to emphasize the transient performance ofthetwopolicies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.4 Expected time average total queue length in the sensor network against the exogenousarrival rates, changing fromλ =1 toλ =10, withthe dashed lines representinginterpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5 Timeslotevolutionoftotalnumberofpacketsinthemulticlassnetworkforthree arrival rates ofλ =1 (top),λ =5 (middle) andλ =10 (bottom) for each class ateachnode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Expected timeaveragetotal queue lengthin themulticlass networkagainst the exogenousarrival rates, changing fromλ =1 toλ =10, withthe dashed lines representinginterpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 A capacitated directed edge is modeled by a nonlinear resistor in series with an idealdiode: (left)Current-voltagecurveofthenonlinearresistorwhichsaturates current atµ ij and shows linear conductivity ofσ ij prior to saturation. (middle) Resistive-voltagecurveofthenonlinearresistor. (right)Current-voltagecurve ofthenonlinearresistorinserieswiththeidealdiode. . . . . . . . . . . . . 128 Listoffigures xvii 5.2 Nodesinjectstwotypesofelectricalchargewithintensity3,onedestinedfor nodeaandanotherfornodeb. Allnonlinearresistorsareofunitconductance and of capacity 5. Electrical currents through the edges follow the multi-charge nonlinearOhmlaw(5.31). . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Geometrical interpretation of solving optimization problem (5.38) for a two- class case. The figures on top are examples with the channel capacity excess e < 0 and on bottom with e > 0. The top-right example illustrates that the optimalintegersolutionisnotnecessarilyunique,astherecouldexistmorethan onevertexoftheunithypercube(squarehere)withequalshortestdistancefrom theinitialuniquenon-integersolution. . . . . . . . . . . . . . . . . . . . . . 140 5.4 Timeslot evolution of total routing cost and total number of packets in the multiclassnetwork,displayingtheperformanceofDirichlet-basedpolicyversus V-parameterBPpolicywithV =0.8underthreearrivalratesofλ =1(left), λ = 5 (middle) and λ = 10 (right) for each class at each node. Besides lowerroutingpenalty,notethefarsmalleroscillationsundertheDirichlet-based routing,whichimpliesasmootherflowoftraffic. Totalnumberofpacketsin the network is the representative of routing delay, where the Dirichlet-based policymakesthebestpossibletrade-offoncostversusdelay. . . . . . . . . . 154 5.5 Expected time average performance of the Dirichlet-based policy versus V- parameterBPpolicywithV =0.8underarrivalratesgrowingfromλ =1to λ =10. Long-termaverageroutingcost(top). Long-termaverageroutingdelay (bottom). Dashedlinesdisplaythirddegreepolynomialinterpolation. . . . . 156 6.1 Intrinsic curvature measured by triangulation. A geodesic triangle is the region bounded by three geodesics that meet in three points. Despite the Euclidean space (flat curvature), in the spherical (positive curvature) and hyperbolic (neg- ative curvature) spaces, the triangles cannot be made from a flat piece of paper without tearing or stretching. By the local Gauss-Bonnet Theorem, the integral curvaturemayberegardedasthecorrectiontoπ ofthesumofinteriorangles. 168 6.2 CirclepackingonatriangulationplanargraphinE 2 wherethelengthmetricℓ isdeterminedfromacombinationofcircleradiiandintersectingangles. . . 172 xviii Listoffigures 6.3 Gromov’s thin triangle condition for a hyperbolic geodesic metric space: (left) Each geodesic is withinδ -neighborhood of the union of the other two. (middle) ThelocalcombinatorialcurvatureateachnodeofX 3,7 hyperbolictessellation isnegative. (right)Arandomgraphformedbyjoiningnodesthatare withina specifiedhyperbolicdistance. . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.4 Traffic on Euclidean space (left) and hyperbolic space (right). The optimal paths (geodesics) are uniformlydistributedina Euclideanspace, butmaximally distributed at the center in a Poincaré space. The intuition is that, because the geodesics in Poincaré disc are arched, they spend more time in the small hyperbolicballthaninthesmallEuclideanball. . . . . . . . . . . . . . . . 178 7.1 Transmissionlinemodelinpowergrids: (left)Transformertapsettingrepresen- tation. (right)TransformerΠ -model. . . . . . . . . . . . . . . . . . . . . . 183 7.2 Node congestion in data networks versus line overload in power grids. In a data network, limitation is on routers with packet drops, whereas in power gridlimitationisontransmissionlineswithoverloadtrips. Inadatanetwork both send/receive and congestion occur in nodes, whereas in power grid sup- ply/demandoccursinbuses(nodes)butoverloadinghappensinlines(links). 195 7.3 Electrical inertia analysis for power stations and transmission lines in the IEEE300bussystem. Transmissionpowercomputedfrompowerflowequations in green versus the inverse of inertia in blue, showing opposite relationship between transmission power and inertia. The following facts are concluded: (1)Thegridislocallynegativelycurvedwithrespecttosomelinesandbuses. (2) Line number 1 has zero inertia and solely connecting the network to the swingbus. (3)Linesnumber0to50areinhighcentralitywithrespecttothe fluctuationsofpowersupply/demandinbuses. . . . . . . . . . . . . . . . . 198 7.4 Effective resistance (electrical distance) curvature analysis in the IEEE300 bus system. The value ofR(a,i)/R eff (a,i) for five sample nodes a in virtual resistive network, where i changes among all other nodes. By Th. 7.3, the flatratioisasymptomoflocal negativecurvatureforsometransmissionlines connectedtothecorrespondingnodea. . . . . . . . . . . . . . . . . . . . . 199 7.5 Overallarchitectureoftopologycontrolinasmartpowergrid,displayingthe required interaction with other control and operation components of power system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.6 Star-DeltatransformationstakingK 5 toaplanargraph. . . . . . . . . . . . . 202 Listoffigures xix 8.1 ArchitectureofaTSOvoltagecontrolagent. . . . . . . . . . . . . . . . . . . . 211 8.2 Normal Mode Operation based on Distributed Model Predictive Control ap- proach. Inasteady-statepractice,eachTSOusesageneraldynamicmodelof its own area as well as a reduced-order QSS model of its neighbors, exchanged ateachtime-slot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Chapter1 Introduction Thebroadinterestofthisresearchistoopenawaytowardsanalysis,controlandoptimization of complex networked systems by proposing a framework that brings three research areas under one envelope, that are, combinatorial geometry, complex networks and stochastic resource optimization. We believe these very active research areas, despite being developed largely separate,couldhighlybenefitfromeachother. 1.1 CombinatorialGeometry Thepracticalimportanceofclassicaldifferentialgeometry stemsfromthefactthattheanalytical dynamicsofmostphysicalphenomenainvolvemultidimensionalpartialdifferentialequations, which may be expressed in terms of exterior differential forms, first introduced by Cartan in 1945. Theessentialpropertythenisthecoordinate-freecalculusofexteriorformsthatprovidesa mathematicalframeworktoexpressdifferentialandintegralequationsonsmoothmanifoldswith the same computational simplicity and concreteness as vector calculus. For instance, the main operationhere,calledexteriorderivative,replacesandgeneralizestheconventionalgradient, divergenceandcurloperationsaswell astheclassicalGreen,GaussandStokestheoremsina consistentandsystematicmanner. Interestingly,combinatorialgeometryprovidesusonatopologicallevelwithverysimilar counterparts of fundamental operators in classical differential geometry. The key insight is that instead of merely pointwise evaluation of all physical quantities with different properties (think of flux for magnetic field, current for electricity and pressure for atoms’ collisions), one can correspondquantitiestovertices,edges,facesandtetrahedraaspropercombinatorialversions 2 Introduction ofrespectivelypointwisefunctions,lineintegrals,surfaceintegralsandvolumeintegrals. For instance, on a piecewise-linear triangle mesh, the Gaussian curvature can be easily proven to be zero everywhere except on vertices, while the mean curvature can only be defined on edges. In this sense, the Gaussian curvature behaves like voltages at the nodes in electrical circuits, while meancurvaturebehaveslikecurrentsalongthebranches. There is a vast body of literature in both theoretical foundations and pragmatic applications of combinatorial geometry in the forms of discrete exterior calculus, mimetic discretization, finite element exterior calculus, computational electromagnetism, and discrete differential geometry. We refer the interested reader to [1–24] and references therein. Much of the progress inthisdirectionhasbeenmotivatedbymakingatransparentrelationshipamongtopological, geometricalanddifferentialstructuresofvariousproblemsintheoreticalphysicssuchasgeneral relativity,electromagnetism,gaugetheory,aswellassolidandfluidmechanics. 1.2 ComplexNetworks Asignificantamountofresearchhasbeendoneonestimatingdifferentstructuralpropertiesof “complexnetworks”arising froma varietyofdatasetsinsocial,informationalandbiological systems. Most noticeable in the literature are “small world” and “scale free” networks [25–29], whichstemfromscientistsdealingwithsocialnetworksthatlackthehomogeneityofdegree distributioninregularandrandomnetworks. Inascale-freenetwork,nodedegreedistributionfollowsapowerlaw,atleastasymptotically. Hence, there is a small group of nodes with a degree that greatly exceeds the average on the network,whichindicatesaheavy-tailedtopographicalconnectivity. Inasmall-worldnetwork,on theotherhand,thetypicaldistancebetweentworandomlychosennodesgrowsproportionallyto thelogarithmofthenumberofnodesinthenetwork. Thiscreatesafat-taileddegreedistribution whereafairnumberofnodeshaveveryhighdegrees. Hence,whilemostnodesarenotneighbors of one another, contrary to that in scale free networks, they can be reached from every other nodebyasmallnumberofhopsorsteps. Though interesting and important, these concepts fail to identify some crucial and intrinsic geometrical and topological characteristics of many new and emerging large-scale networks. The reasons, besides the size, are that firstly, the philosophy of connection and commodity traversalhereisverydifferentfromthatinsocialnetworks,andsecondly,thesenetworkshave highlytime-varyingtopologiesinthescaleofhoursorminutes. Thesefeatureschallengefor additionalstructuralpropertiesinunderstandingandestimatingnetworkperformance,reliability 1.3StochasticResourceOptimization 3 andsecurityinanefficientandscalablemanner. Apromisingapproachisthatof graphcurvature adopted from Riemannian geometry [30–39]. In this spirit, the graph of a network is viewed as coarselyequivalenttoaRiemannianmanifoldand,assuch,thecrucialnetworkcharacteristic turnsto beitscurvature. To make the concept of graph curvature more palatable, it may be said that a (highly connected,meshed)small-worldnetworkispositivelycurvedwhilea(core-concentric,tree-like) scale-freenetworkisnegativelycurved. Butthisidentificationistenuous[ 31],whichisprecisely thesubtlediscrepancybetweentheoldconceptandtheneweronethatmakesitsospecific. Most importantly, the notion of coarse geometry approach has allowed the explanation of congestion in the internet backbone and communication networks, referring to the strong concentration of trafficonsomesmallsubsetsoflinks/nodes. Thisiswhilethelocalheavy-tailedparadigmof scale-freenetworksfailstoexplainthisphenomenon. 1.3 StochasticResource Optimization LetanetworkrepresentasetofserversE andasetofplayersV,whereeachserverℓ∈E can serveatamaximumrateofµ ℓ >0 andeachplayeri∈V generatesautilityvalueofU i (x i ) in receiving the service rate ofx i > 0. LetK(ℓ)∈V represent set of players that need service fromserverℓ. Ageneralutilityoptimizationframeworkisformulatedas max x i X i∈V U i (x i ) s.t. P i∈K(ℓ) x i 6µ ℓ (1.1) where the constraint ensures that server capacities are not violated. Objective functions can takevariousnetworkparametersintoaccountsuchastrafficrate,latency,transmissionpower andjitter,andcanmodeluserbehavior,operatorcostortrafficelasticity. Theycanalsodefine the fairness of resource allocation in different forms such as α -fairness, max-min fairness, proportionalfairnessandthroughputmaximization. By distributing (1.1) throughout the network, decentralized algorithms can be developed to control rate allocation based only on local variables such as server price and load, where convex programming and Lagrange duality are the key tools. By decomposing (1.1) to different sub-problems, on the other hand, theoriginal problem canbe solved throughdifferent network layers. Suchacross-layeroptimizationframework[40]hasledtojointcontrolofcongestion, routing, scheduling,transmissionpower,channelcoding,andsoon. 4 Introduction Manynetworkedsystemssharesimilarcommonobjectives,including: (Stability) Prior to any other performance criteria, a network needs to remain stable, where stabilityusuallyreferstoboundednetworkdelay,lossprobabilityorbacklog. (Complexity) A network control policy needs to have reasonable complexity from both as- pectsofcomputationandimplementation. Thisalsobearsthenotionsofefficiencyand scalabilitywithrespecttothenetworksizeandresources. (Robustness) Anetworkneedstoberesilienttounpredictablechangesintopologyandenvi- ronment,wherebasiccriteriaaretosustainthenetworkstabilityandfunctionality. (Optimization) Limited network resources need to allocate among competing players in a way to optimize a network-wide objective function in the form of utility maximization or penalty minimization. A commonly used measure of network-wide utility, or penalty for thatmatter,isthesumoftheutilitiesorpenaltiesofalltheplayers. In the presence of random uncertainties in traffic (supply/demand) and topology (structure), optimalresourceallocationturnstobeastochastictime-varyingparameterproblemthatisoften challengingtosolveforeithernetworkstabilityorsolutionfeasibility,i.e.,honoringnetwork constraints. Further, in such complicated problems as cross-layer optimization in wireless networksandoptimalpowerflowinpowergrids,utilityfunctionsarefrequentlynon-convex [41,42]. Hence,inadditiontodeterministicconvexprogramming,othertechniquesareoften requiredtosolvenetworkutilityoptimizationproblems. 1.4 CommunicationNetworks versus Power Grids Among different complex networks, we focus on communication networks and power grids. Thesenetworked systemsshape twocritical infrastructuresof modernsociety, whileboth face newsubtleproblemswhosesolutionsrequiremuchinnovationandresearch. Communicationnetworks are embracingthe rapidgrowthof wirelessand adhocnetworks thatarechallengingduetotheintrinsictime-varyingfeaturesofwirelessconnectivity,including multi-path propagation and fading, limited energy (battery), user mobility and inter-channel interference. Powergridsarechallengedbytherecentandongoingmarketderegulation,theneed ofintegratingagreatamountofintermittentrenewableenergygeneration,andtheemerging conceptofdata-driven“smartgrid”. 1.5ThesisOutline 5 Communication networks and power grids are both large-scale, which makes them ideal frameworksforcapturingtheintricaciesofcomplexnetworks. Whenitcomestopowergrids, the complexity is not only in the network topology but also in the network interconnection dynamics. Further, both communication networks and power grids share some other important network characteristicssuchassparsity,passivityandstochasticity. Communication networks and power grids are called for the convergence towards smart grids that gather, distribute and act on information about the behavior of power system in order to improvethe efficiency, competitiveness,reliability and security ofboth communication and powernetworks. Thissupposesaradicalchangeinthewayenergyandinformationisgenerated, distributed and consumed, and accordingly specifies the technical challenges in the design and managementofthesetwocomplexnetworkedsystems. Lastbutnotleast,communicationnetworksandpowergridsarerepresentativesoftwomain classesofnetworkedsystemswith thefollowingspecifications: • Protocol-driventransportphenomena: These include wireless networks, manufacturing systemsandhighwaytransportation(consideringtrafficregulationsasimplementinga human-designedprotocol). • Physics-driven transport phenomena: These refer to power grids where power flows according to the power flow equations, gasand liquiddistributionthat complywithfluid mechanics, and spin networks where the excitation from one spin to another follows a probabilityrulederivedfromtheFeynmanpathintegral. In thefirst subsetof phenomena,traffic is fullycontrollable inthe sense thata humanoperator could change the protocol as it fits. Hence, once the cause of congestion is identified, it is relatively straightforward to develop a cure: in wireless networks, for instance, it suffices to redefinethelinkweightssoastomakethenetworkpositivelycurved[ 35, 36]. Inthesecond subset, control authority is very limited, as the traffic is not driven through the network by a man-madesystem,butfollowsthelawsofnature. However,withthenotionofcyber-physical systems,ithasbecomepossibletoimplementcontrolontopofthelawsofnaturesothatnatural phenomenaevolveinthewaythehumanoperatorwants. 1.5 ThesisOutline Thistextisdividedintotwoparts: PartI:Wirelessnetworks,andPartII:Powergrids. 6 Introduction Part I is composed of four chapters. In Ch. 2 we briefly review the notion of combinatorial geometry that equips us with basic theory and tools to derive combinatorial counterparts of fundamentaloperatorsin differentialgeometry. Wewillseehowthesystematicdescriptionof cellcomplexasadiscretedomainanditsassociatedoperatorsallowustoformulatedifferent combinatorialoperationsintheformofmatrixequations. In Ch. 3 we propose a dynamic routing protocol, referred to as Heat-Diffusion (HD), for uniclass wireless networks operated under random traffic, time-varying topology and inter- channel interference. The major result is to show that HD is Pareto optimal with respect to average network delay and average quadratic routing cost, where the level of compromise is managed by a routing parameter 0 6 β 6 1 that trades network delay for routing cost. The protocol is named Heat Diffusion as its Dirichlet energy criterion is derived from the combinatorial analogue of the classical heat equation on smooth manifolds by translating “heat quantitymeasuredincalories”to“queueoccupancymeasuredinpackets.” Topreserveimportantfeaturesofheatpropagation,includingtheprincipleofenergymin- imization, we exploit tools and techniques from combinatorial geometry, which allow us to directlytransferprinciplesofphysicstoacombinatorialsetting. Indoingso,weencountered thefollowingmajorresearchchallenge: Notonlydoweneedamanifoldtographtranslation, but a further translation from undirected graphs (Riemannian manifolds) to directed graphs (Finslermanifolds)toaccommodatethedirectionalityofedges,fromuncapacitatedgraphsto capacitated graphs to accommodate the limited transmission capacity of edges, and from single commoditytomultiplecommoditiestoaccommodatethemulticlassnatureofwirelessnetworks. ThefirsttwochallengesareaddressedinCh.3andthesolutionisextendedtosupportlimited capacityandmultiplecommoditiesinCh.5. InCh.4weextend theresultofHDprotocoltomulticlassnetworksforthespecialcaseof β =0,where thegoalisto minimizeaveragenetworkdelay. Intheory, achievingthisgoalfor a general case requires not only arrival statistics, channel state probabilities and the Markov structure oftopologyprocess,butalso solving a dynamicprogram for each possible topology, which make it unreachable in practice. However, we propose a control policy that achieves this goalintheclassofallroutingpoliciesthatmakecontroldecisionbasedonlyoncurrentcondition of queue occupancies and channel states, including those with perfect probability knowledge of trafficandtopology. Thepointisthatourapproachdoesnotrequireanyinformationabouttraffic andtopologyanddoesnotdealwithanydynamicprogramingorMarkovdecisionprocess. In Ch. 5 we first propose the notion of nonlinear multi-charge conduction current on capaci- tated directed graphs and establish a one-to-one mapping between that and multiclass wireless 1.5ThesisOutline 7 networks. Usingthis,wethenextendtheresultofHDprotocoltomulticlassnetworksforthe specialcaseofβ =1,wherethegoalistominimizeaveragequadraticroutingcostinthesense of Dirichlet. Themajor resultis to show that the fluid limit of interference multiclass packet flowmimicsasuitably-weightednon-interferencemulti-chargeconductioncurrent,whichcan openanewwayintheanalysisandoptimizationofstochasticprocessingnetworks. PartIIbeginswithCh.2thatoverviewsthenotionofgraphcurvatureinthesenseofGromov asastructuralmetricforweightedgraphs. Theaimistounderstandtheimpactofunderlying graphcurvatureonsuchfundamentalphenomenaovercomplexnetworksastrafficcongestion. Wethenextendtheconceptofcurvaturetoresistivenetworks,whereanelectricalanalogueof curvatureisintroducedbyusingeffectiveresistanceasanelectricaldistance. InCh.7weaddresstheimpactoftopology,inthesenseofcurvature,onthemonitoringand control of electric power grids. The specific objective is to study the influence of grid topology on distribution of network sensitivity with respect to supply/demand fluctuations. Such a study can associate the structure of power network with the traditional practice of sensitivity analysis in power community. It can also shed a light on the security of power system as a complex network,whichisimportantinanalysisofsystemvulnerabilitytoelectricalblackoutsderived bycascadingfailures,orinformationblackoutsderivedbycyberattacks. Part II ends in Ch. 8 which is devoted to the voltage stability and control of multi-area powernetworks,definedasaninterconnectionofseveralpowersystemsoperatedbydifferent independent entities without centralized coordination. This is an arising topic due to electricity marketderegulation,substantialadditionofrenewableenergygenerationthatisvariableand uncertain,andrecentlargeanddevastatingblackoutsinNorthAmericaandEurope. Combining theories and techniques from distributed control and mathematical programming, the goal is to coordinate the control actions among neighboring grids while preserving sensitive local system datathatregionaloperatorsareoftennotwillingtodisclose. PartI WirelessNetworks Chapter2 SmoothversusCombinatorialGeometry This chapter briefly compares combinatorial geometry with classical smooth geometry. Our goalistodemonstratehowthestructureandfundamentaloperatorsincontinuousdifferential geometry find similar counterparts in combinatorial geometry. Starting with the concept of cochainsasthecombinatorialcounterpartofdifferentialforms,wethenproceedtothedefinition of coboundary operator as the combinatorial version of exterior derivative in smooth geometry. Introducing dual complex leads to combinatorial Hodge operator to map cochains from primal complex to dual complex, scaled by the inverse metric tensor. The Hodge operator opens the way to define the combinatorial analogue of codifferential operator in smooth geometry. With combinatorial coboundary and codifferential operators in place, the culmination is the combinatorialversionoftheLaplace-deRhamoperatorthatisfrequentlyusedthroughoutthe restofthisresearch. Incontrasttospacediscretization,wherethemaingoalistofindanaccuratetriangulation of a manifold to serve as the computational grid, here we deal with a discrete domain, such as a graph, entirely as its own entity with no reference to any underlying continuous process. Inother words,while numerical partialdifferentialequationsputemphasisonthefidelityofa discreteapproximationtothedesiredanalyticalsolution,combinatorialgeometryestablishes aseparate,equivalentframeworkthatoperatesonapurediscretedomainfrombeginning,of which a graph is a special case that includes only zero and one complexes. Consequently, in the context of combinatorial geometry, the concerns of numerical discretization about approaching acontinuoussolutioninthelimitareirrelevant. We remark that this intrinsic difference between space discretization and combinatorial geometrydoesnotmakeanyoftheseframeworksinferiortotheotherandbothprovidepowerful techniquestodescribeandsolvedifferentphysicalandcomputationalproblems. Forinstance, 12 SmoothversusCombinatorialGeometry while the space discretization of Maxwell’s equations has led to the so-called “discrete elec- tromagnetism,”celebratedKirchhoff’sequationsinelectricalcircuitsarebestdescribedinthe framework ofcombinatorialgeometry. 2.1 SmoothGeometry Since the main objective of combinatorial geometry is to provide discrete analogies for certain elements and operators in smooth differential geometry, it will be useful to first collect and brieflydescribethoseelementsandoperatorsincontinuousdomain. 2.1.1 Topologicalmanifolds TwotopologicalspacesX andY arehomomorphic,i.e.,essentiallyequivalentfromatopological perspective,if thereis abijectionφ :X →Y forwhich bothφ andφ − 1 are continuous. Then φ is called a homomorphism betweenX andY. A mapφ : X → Y is an embedding ifφ yields a homomorphism betweenX andφ (X), whereφ (X) carries the subspace topology inherited fromY. Every embedding is injective and continuous and every map that is injective, continuousandeitheropenorclosedmakesanembedding. An n-manifoldM is a topological subspace that is locally equivalent toR n , i.e., every point x ∈ M has an open neighborhoodU homomorphic to an open subset ofR n . A manifold with a boundary can be defined similarly, except that every point has an open neighborhood homomorphictoanopensubsetoftheupperhalfplaneinR n . A covering space of manifoldM is a manifoldN together with a local homomorphism φ :N →M such that each point onM has a neighborhood that is evenly covered byφ . Thus, for everyx∈M there exists an open neighborhoodU such thatφ − 1 (U) is a disjoint union of open sets inM. A covering space is called universal cover if it is simply connected. The universalcoverofM coversallcoveringspacesofM. A coordinate chart, or a system of local coordinates, is a homomorphismφ :U →φ (U)∈ R n withU ⊂ M being an open set. If (U,φ ) and (V,ψ ) are two charts with overlap, the homomorphismφ ◦ ψ − 1 : ψ (U∩V)→ φ (U∩V), as a map between two open sets ofR n , iscalledthetransitionmapfromψ toφ . TwochartsareC k -compatibleifthetransitionmap between them isC k -diffeomorphic. A family of C k -compatible charts is called aC k -atlas if their domains cover the whole manifold. AC k -manifold is then defined as a topological manifold endowed with aC k -atlas, whereC ∞ -manifolds are also called smooth manifolds. 2.1SmoothGeometry 13 Looking at compatibility as an equivalence relation, a differentiable structure for a smooth manifoldisdefinedasanequivalenceclassofsmoothatlases. 2.1.2 Tangentandcotangentspaces Given a smoothn-manifoldM, letx 1 ,··· ,x n be a system of local coordinates in a chartU. The tangent space ofM at a pointx, denoted byT x M, is the collection of all tangent vectors atx. The tangent space does not depend on the choice of coordinate chart. The tangent bundle ofM, denoted byTM, is then defined as the disjoint union of tangent spaces at all points, TM= S x∈M T x M. Fix a pointx∈M and letf be a smooth function in a neighborhood ofx. Thinking of ξ ∈T x M asadirectionatpointx,thenotionofdirectionalderivativecanbedefinedby ξ (f):=∂f(x)/∂ξ = ξ i ∂f(x)/∂x i n i=1 ∈R n . Thenwedefinethe differential off asalinearfunctionalonT x M suchthat df(x),ξ :=ξ (f), ∀ξ ∈T x M where [·,·] represents the pairing between a linear functional onT x M and a tangent vector. Hence,df(x) isanelementofthedual spaceT ∗ x M, which is calledthe cotangent spaceand itselementsarecalledcovectors. Thedisjointunionofcotangentspacesatallpointsiscalled cotangentbundleT ∗ M= S x∈M T ∗ x M. All the partial derivatives∂/∂x i evaluated atx are linearly independent, which implies that {∂/∂x i } n i=1 is a basis in the linear spaceT x M with dim(T x M) = n. Hence, any tangent vectorξ ∈T x Mcanberepresentedintheformofξ = P n i=1 ξ i ∂/∂x i withξ i =ξ (x i )being a component of the vectorξ in a local coordinate chart. Since the dual space is also a linear spaceofthesamedimension,anybasis{e 1 ,··· ,e n }inT x Mhasadualbasis{e 1 ,··· ,e n } inT ∗ x M suchthat[e i ,e j ]=δ i j ,whereδ i j istheKroneckerdelta. The covectordf(x) can be represented bydf(x) = P n i=1 (∂f(x)/∂x i )dx i in the basis {dx i } n i=1 . Thisimpliesthatthepartialderivatives∂f(x)/∂x j areindeedthecomponentsofthe differential df(x) inthebasis{dx i } n i=1 . Therefore,foranyj =1,··· ,n,weget ∂f(x) ∂x i dx i , ∂ ∂x j = ∂f(x) ∂x i dx i , ∂ ∂x j = df(x), ∂ ∂x j . 14 SmoothversusCombinatorialGeometry 2.1.3 Metrictensorandinnerproduct A Riemannianmetric onasmooth manifoldM,denoted byg M ={g(x)} x∈M ,is afamilyof smoothlyvarying,symmetric,positivedefiniteandbilinearformsonthetangentspacesof M. A Riemannian manifold is then defined by a pair (M,g M ). Locally, a metric can be described in terms of its coefficients in a local chart, defined by g(x)=(g ij ) n i,j=1 ∈R n× n . The functions g ij are called the components ofg(x) in the local chart. The smoothness ofg(x) entails every componentg ij tobeaC ∞ -functioninitscorrespondingchart. Usingthemetrictensor,onecandefinean innerproduct ineverytangentspaceby ξ ,η g :=g(x)(ξ ,η )=g(ξ ,η ), ∀ξ ,η ∈T x M. Thenwecanimmediatelydefinethe normofavectorξ by∥ξ ∥=⟨ξ ,ξ ⟩ 1/2 g ,whichalsoreads thelengthofξ . Inalocalcoordinatechart,weobtain ξ ,η g = X i,j g ij ξ i η j whichimpliesg ij =⟨∂/∂x i ,∂/∂x j ⟩ g . For every vectorξ ∈ T ∗ x M, there exists a corresponding vectorg(x)ξ ∈ T ∗ x M that is defined by the identity ⟨g(x)ξ ,η ⟩ =⟨ξ ,η ⟩ g ,∀η ∈ T x M. Hence, we getg(x) : T x M→ T ∗ x M,whichreadsanisomorphisminducedbythemetricg. Further,thereexiststheinverse mappingg − 1 (x) : T ∗ x M→ T x M, whose componentsg ij are defined by g ij = (g − 1 (x)) ij . Similarly,g − 1 (x) definesaninnerproductin T ∗ x M,suchthat v,ω g − 1 = g − 1 (x)v,g − 1 (x)ω g , ∀v,ω∈T ∗ x M. 2.1.4 Pathsandgeodesics A parametric path inM is a continuous mapγ :[a,b]→M, where[a,b]⊂ R. A pathγ (t) is definedbyitscomponents γ i (t)inalocalchart,andiscalledsmoothifallitscomponentsare smooth functions of parametert. Given a smooth pathγ :[a,b]→M with the velocity vector γ ′ (t),aRiemannianmetricprovidesadefinitionofthe length ofγ (t) by l(γ ):= Z b a ∥γ ′ (t)∥dt. 2.1SmoothGeometry 15 In a local chart containing the image ofγ (t), we get∥γ ′ (t)∥ 2 = P i,j g ij (γ ′ (t))γ ′i (t)γ ′j (t). It is then immediate that every smooth path can be reparametrized as a unit-speed path, i.e., one forwhichthevelocityvectorhasunitlengthwithrespecttotheRiemannianmetric. We say that a smooth pathγ :[a,b]→M connects pointsx andy onM ifγ (a)=x and γ (b)=y. Thenthegeodesicdistanced(x,y) isdefinedby d(x,y):=inf γ l(γ ) where the infimum is taken over all smooth paths connecting the points x andy. If there exists noconnectingpathbetweenxandy,thend(x,y)=∞. Thepathcorrespondedtothegeodesic distance iscalledtheshortestpathorgeodesic. ItisproventhatthegeodesicdistanceisametriconMandthetopologyofthemetricspace (M,d) coincideswiththetopologyoftheoriginalsmoothmanifoldM. 2.1.5 Riemannianmeasure Given a smooth manifold M, a set E ⊂ M is called measurable if for any chart U, the intersectionE∩U isaLebesguemeasurablesetinU. ThefamilyofallmeasurablesetsinM forms aσ -algebraΛ( M). It is proven that for any Riemannian manifold(M,g), there exists a uniquemeasureϑ onΛ( M) suchthatinanychartU, dϑ= p detg(x)dλ withλ beingtheLebesguemeasureinU. ItfollowsthatforanymeasurablesetA⊂ Λ( M), ϑ(A)= Z A p detg(x)dλ. Further, the measure ϑ is complete, meaning that ϑ(K) < ∞ for any compact setK ⊂ M ; ϑ(N)>0 foranynon-emptyopensetN ⊂M ;andϑ isregularinthesensethat ϑ(A)=sup ϑ(K):K⊂A , K compact =inf ϑ(U):A⊂N , N open . Themeasureϑisextendedfromchartstothewholemanifoldandsinceϑisfiniteoncompact sets,anycontinuousfunctiononM isintegrablewithrespecttoϑ. 16 SmoothversusCombinatorialGeometry 2.1.6 Gradient,divergenceandLaplaceoperators Let(M,g M ) be a Riemannian manifold and letf be a smooth function in a neighborhood of a pointx∈M. Thegradientoff atx isdefinedby ∇f(x):=g − 1 (x)df(x). Thus,weobtainthecomponentsofthegradientinalocalcoordinatechartas ∇ i =g ij ∂ ∂x j where, for the special case ofg being the identity matrix, retrieves the classical gradient in the Euclidean space. Itimmediatelyfollowsthat∇f(x) isacovectorofdf(x),whichimplies ∇f(x),ξ g = df(x),g(x)ξ g − 1 , ∀ξ ∈T x M. Thisprovidesanalternativedefinitionofthegradient. Inparticular, with hasanothersmooth functioninaneighborhoodofx∈M,thefollowingidentitieshold ∇f(x),∇h(x) g = df(x),dh(x) g − 1 = X i,j g ij ∂f(x) ∂x i ∂h(x) ∂x j . A vector field onM, denoted by{v(x)} x∈M , is defined as a family of tangent vectors v(x)∈T x M foranyx∈M. Usingv(x)=v i ∂/∂x i inalocalchart,v(x) iscalledsmooth ifitseverycomponentv i isasmoothfunctioninanychart. Forasmoothvectorfield v(x)on M,itsdivergenceisasmoothfunctiononM suchthat Z M divv(x) f(x)dV := Z M v(x),∇f(x) g dV , ∀f(x)∈C ∞ (M) wheredV istheinfinitesimalmeasuringelementin M. Inalocalcoordinatechart,thedivergence ofavectorfield v canbedeterminedby divv(x)= X i 1 p detg(x) ∂ ∂x i p detg(x)v i = X i ∂v i ∂x i +v i ∂ ∂x i log p detg(x) . 2.1SmoothGeometry 17 For the special case ofdetg(x)=1, we obtaindivv(x)=∂v i /∂x i , which reads the classical divergenceintheEuclideanspace. The Laplace-Beltrami operator is defined as the divergence of the gradient. Thus, the Laplacianofasmoothfunctionf inaneighborhoodofx∈M isdefinedby ∆ f(x):=div ∇f(x) . Bythedefinitionofdivergence, ∆ f(x) isasmoothfunctiontoo. Inalocalchart,weobtain ∆= X i,j 1 p detg(x) ∂ ∂x i p detg(x)g ij ∂ ∂x j . Forthespecialcaseofg(x)beingtheidentitymatrix,weobtain∆= ∂ 2 /(∂x i ) 2 ,whichretrieves theclassicalLaplacianintheEuclideanspace. Lettingf(x) andh(x) besmoothfunctionson(M,g M ),theGreen’sformula reads Z M f(x)∆ h(x)dV =− Z M ∇f(x),∇h(x) g dV = Z M h(x)∆ f(x)dV. 2.1.7 Weightedmanifolds AnysmoothpositivefunctionD(x)on(M,g M )inducesameasureµ onMgivenbydµ =Ddϑ, whereD iscalledthedensityfunctionofthemeasureµ . Forinstance,thedensityfunctionof theRiemannianmeasureϑis1. Wedefinea weightedmanifoldbyatriple(M,g M ,µ )whereµ isameasurewithasmoothpositivedensityfunctionon(M,g M ). The gradientremains the same ona weighted manifold. However,the weighted divergence ofasmoothvectorfield v(x) on(M,g M ,µ ) isdefinedby div µ v(x):= div D(x)v(x) D(x) . Itfollowsfromthedefinitionofdivergencethat Z M div µ v(x)f(x)dµ =− Z M v(x),∇f(x) g dµ, ∀f(x)∈C ∞ (M). TheweightedLaplace-Beltrami operatoristhendefinedas ∆ µ f(x):=div µ ∇f(x) . 18 SmoothversusCombinatorialGeometry Defining ρ (x):=D(x) p detg(x),inalocalcoordinatechartU,weobtain div µ v(x)= 1 ρ (x) ∂ ∂x i ρ (x)v i , ∆ µ f(x)= 1 ρ (x) ∂ ∂x i ρ (x)g ij ∂f(x) ∂x j . Notealsothatdµ =ρ (x)dλ withλ beingtheLebesguemeasureinU. For two smooth functionsf(x) andg(x) on a weighted manifold(M,g M ,µ ), the Green’s formularemainsunchanged,thatis, Z M f(x)∆ µ h(x)dµ =− Z M ∇f(x),∇h(x) g dµ = Z M h(x)∆ µ f(x)dµ. 2.2 CombinatorialGeometry Combinatorial geometry interestingly provides us on a topological level with very similar counterpartsoffundamentaloperatorsfromsmoothgeometry. Theconnectionbetweenthese two is made possible by linking the spaces of their respective main objects of interest, i.e., differentialforms on a manifold and cochainsonacell complex. Theobjective ofthischapter is to present combinatorial analogies to the standard tools used in smooth geometry. The correspondingtools find theirorigins inthefieldof algebraictopology, andmorespecificallyin computational homology and cohomology, where their combinatorial nature turns the resulting operationsintosimplematrixmultiplications. Formoreexplanationsanddetaileddiscussion, wereferthereaderto[43]andreferencestherein. 2.2.1 Cellcomplex In combinatorial geometry, we work with a cell complex as a countable discrete domain, which just so happens to be the main object of interest in algebraic topology. A cell complex is a collectionoffinite-dimensionalvectorspacesofbasicbuildingblocks, called cells. Ak-cell σ k isdefinedas asetof K>k geometricallyindependentverticeshomeomorphictoaclosed k-ball. The boundary of ak-cell is part of the cell that is mapped to the boundary of the ball by thehomeomorphism. Ann-dimensional cell complex is the natural analogy of a piecewise flat n -manifold, which is a cell complex with cells drawn on the manifold. We can think of it as cells “glued to” a smooth manifold in such a way that they form a curved cell complex. A piecewise flat manifold isageometricmanifoldinthesensethatitisendowedwithapiecewiselinearstructureinmuch 2.2CombinatorialGeometry 19 thesamewayasaRiemannianmanifoldbeingendowedwithadifferentiablelinearstructure. Similarly, we can define a length function to calculate the distance by minimizing the lengths of piecewiselinearcurves. Thesimplestk-celliscalledk-simplexthatissimplyak-cellwith(k+1)-vertices,represented by[v 0 ,··· ,v k ]. Anyk-simplexrepresentsoneoftwoequivalenceclassesofvertexorderings. Two orderings are called coherent if one is an even permutation of the other. Such a class of ordering is called orientation. A simplex induces an orientation on its boundary simplexes that canbecoherentorincoherentwiththeprimalorientationofaboundarysimplex. AnorientedcellcomplexP ofdimensionn,calledn-complex,isacollectionoforiented cellswhichsatisfytworequirements: (i)everyboundaryelementofeachcellinP isalsoan elementinP,(ii)theintersectionofanytwocellsinP iseitheremptyoranentireboundary elementforbothcells. Ann-complexmaybeorientedthroughtheprocessofsimplicialdecompositionthatsub- divideseverycellintointernalsimplexes. Thenbyorientingonesimplex,theorientationcan propagate to the entire cell. A 0-cell has only one possible ordering, and so it has no classi- cal orientation. However, one may orient a vertex as a source or as a sink, where the source orientationmeansthatthetailofanincidentedgeiscoherentwiththevertex. 2.2.2 Chainsandcochains Givenann-complexP,wedefinea k-skeletonP k ⊂P asthesubsetofallorientedk-cellsin P. We denotethe cardinality ofP k withn k . Thek-skeletonP k is ordered by assigning each k-cellwithapositioninann k -tuple. A k-chainc k is defined as a map c k : P k (P) → R that assigns a scalar value to each orientedk-cell. Chainadditionisdefinedbyaddingupthechainvalues,suchthatreversinga cellorientationchangesthesignofthechainvalueforthatcell. Given an orderedk-skeleton, we define elementary chainσ i k ∈R n k as ann k -vector that takes value 1 for the entry corresponding to thei-thk-cell and 0 for the other entries. Then any k-chaincanbeuniquelyrepresentedasasumofelementarychainsby c k = X n k i=1 c k (σ i k )σ i k , c k (− σ i k )=− c k (σ i k ) wherec k (σ i k ) reads the value ofi-thk-cell inc k and− σ i k represents the inverse orientation forσ i k . Thus,we obtainc k = P i a i σ i k ,a i ∈R. Wedenotethe vectorspace ofallk-chains by C k (P),forwhichthesetofallelementaryk-chains{σ i k }isabasis. 20 SmoothversusCombinatorialGeometry A k-cochain c k is defined as the dual of a k-chain that maps a k-chain c k to a scalar, c k : C k (P) → R. Since a chain is a linear combination of oriented cells, a cochain also correspondstoonevaluepercell,wherec k (− σ i k )=− c k (σ i k ). Usingtheexpressionofc k as a linear combination of elementary chains, we obtainc k ( P i a i σ i k ) = P i a i c k (σ i k ), a i ∈ R. Further,fortwocochainsc k 1 ,c k 2 ∈C k (P),weobtain (c k 1 +c k 2 )(c k )=c k 1 (c k )+c k 2 (c k ). Representing ak-chain as a vector of sizen k , one may represent ak-cochain as a vector ofthesamesize. Theelementarycochainσ k i isdefinedasacochainthattakesvalue1onthe elementary chainσ i k and 0 on the other elementary chains, that is,σ k j (σ i k ) = δ i j . Thus, each cochaincanbeseenasalinearcombinationoftheelementaryk-cochains. ThenthespaceC k (P) can be seen asthe dual space ofC k (P), which is spannedby the set of unitk-cochainsσ k i that constitutesthe basisdual tothestandardbasisofC k (P). Remark 2.1. The importance of cochains is that they can be viewed as the combinatorial analogies to differential forms . Given a Riemannian manifoldM, recall that at any point x∈M, ak-form is a multilinear functional that mapsk-th exterior power of cotangent bundle T ∗ M to a scalar. Thenthe set of all differential k-forms onM defines a vector space Ω k (M). Therefore, a 0-form is a function evaluated at each point, a 1-form is a differential evaluated at each curve, and a 2 or higher order form is an antisymmetric tensor evaluated on each correspondingmanifold. Now,ifwerestrictintegrationtotakeplaceonlyonthek-submanifolds, whichisthesumofk-cellsinP,weclearlyobtainak-cochain. 2.2.3 Metrictensorandinnerproductoncomplex Representingk-chainsandk-cochains byn k -vectors in dual spaces, the linear operation ofa k-cochainc k onak-chainc k readsasaninnerproduct: c k (c k )= c k ,c k . Thenonecandefinea k-metricg k :C k (P)×C k (P)→R withthecomponentsof g k (i,j):= σ i k ,σ j k , 16i,j6n k . 2.2CombinatorialGeometry 21 Since the inner product is nondegenerate,g k is positive definite. Similarly, the dual metric is definedas g k :C k (P)×C k (P)→R withthecomponentsof g k (i,j):= σ k i ,σ k j , 16i,j6n k . Itisaneasypracticetoconfirmthat g k =g − 1 k . Usingthemetrictensorg k ,wecandefineaninnerproductonthespaceof k-chainsas g k (c 1 k ,c 2 k ):= g k c 1 k ,c 2 k , ∀c 1 k ,c 2 k ∈C k (P). Similarly,g k endowsthespaceofk-cochainswithaninnerproductas g k (c k 1 ,c k 2 ):= g k c k 1 ,c k 2 , ∀c k 1 ,c k 2 ∈C k (P). Thus, we getg k (g k c 1 k ,g k c 2 k ) = g k (c 1 k ,c 2 k ), which implies that the mapping obtained byg k preservesinnerproductontheC k (P) space. Bythenotionofmetrictensor,convertingak-chainc k ∈C k intoitscorrespondingk-cochain c k takes on the simple form ofc k =g k c k . Similarly, the dual metricg k =g − 1 k computesk- chains from k-cochains. Therefore, analogous to the Riemannian geometry, where tensorg inducesanisomorphismbetweentangentandcotangentspaces,hereincellgeometry,g k induces anisomorphismbetweenk-chainandk-cochainvectorspaces: g k :C k (P)→C k (P). Contrary to the Riemannian geometry, the basis set of k-chains can always be defined orthogonal,soasg k (i,i)=0,i̸=j. Thissimplifiesthemetrictensortoadiagonalmatrix,at the expense of increasing dimensions in the coarse, sparse topology of a cell complex. Note thatinthespecialcasewhereg k (i,i)=1,16i6n k ,themetric isanalogoustoaEuclidean metricinwhichthereisnodistinctionbetweenchainsandcochains. Analogous toa Riemannian manifold, we define a metrized complex with a pair(P,g P ), whereg P isthecollectionofallk-metricsg k onthecellcomplexP. 2.2.4 Combinatorialexteriorderivative On a Riemannian manifoldM, the exterior derivatived maps k-forms into (k +1)-forms, d : Ω k (M)→ Ω k+1 (M). Atk = 0, where the operand is a smooth function (a 0-form), the 22 SmoothversusCombinatorialGeometry exterior derivative is simplified to the differential operator defined in Sec. 2.1.2. One important propertyisthatd◦ d=0,whichisindeedtheresultofthebasictopologicalfactthat∂◦ ∂ =0, i.e., the boundary of a boundary is empty. A fundamental relationship between the exterior derivative andtheboundaryoperator isstatedbyStokes’theorem, Z M dω = Z ∂M ω whereM isak-manifoldandω isa(k− 1)-forminthecotangentbundleT ∗ M. As a preliminary to defining the combinatorial version of the exterior derivative, let us define combinatorialboundaryoperatorbyhomomorphism∂ k :C k (P)→C k− 1 (P)thatisfirst definedon k-simplexes. Then∂ k isextendedtok-cellsbysimplicialdecomposition,andfinally isappliedtoanyk-chainbytheadditivityassumption, ∂ k X n k i=1 a i σ i k = X n k i=1 a i ∂(σ i k ), a i ∈R. Wedefinetheboundaryofa k-simplexas ∂ k (v 0 ,··· ,v k ) := X k i=1 (− 1) i (v 0 ,··· ,ˆ v i ,··· ,v k ) where(v 0 ,··· ,ˆ v i ,··· ,v k ) denotesthe(k− 1)-simplexobtainedbyremovingthevertexv i . Notethattheboundaryoperatoronak-cellcanbeviewedasaparticulartypeofthek-chain that takes only values of 0, 1 and− 1. Analogous to the smooth geometry, it is an easy practice toconfirmthat ∂ k ◦ ∂ k+1 =0. The combinatorial exterior derivative is defined as the dual of the boundary operator, which is a mapd k : C k− 1 (P) → C k (P) that operates on cochains. Using the property of bilinear pairingbetweencochainsandchains,foranyc k ∈C k andc k+1 ∈C k+1 ,weobtain d k c k ,c k+1 = c k ,∂ k+1 c k+1 . Therefore, the exterior derivative is the adjoint of the boundary operator, that is,d k =adj(∂ k ). Forthisveryreason,d k isalsocalledcoboundaryoperator. Remark 2.2. One can think of integration as the duality pairing between a chain and a cochain. Inotherwords,acochaincanbeviewedasadensityfunctionappliedtothecellsoverachain, whichrepresentsacombinatorialexpressionoftheintegration. Withthisview,wecanregard the definition of d k as a combinatorial version of Stokes’ theorem. More precisely, given a 2.2CombinatorialGeometry 23 (k+1)-chaincandak-cochainω,thecombinatorialStokes’theoremreads⟨dω,c⟩=⟨ω,∂c⟩, whichistruebydefinition. Analogous to the smooth exterior derivative, it follows thatd k+1 ◦ d k =0, simply because theboundaryofaboundaryisempty. 2.2.5 Homologyandcohomology Homologytheoryprovidesawaytoanalyzeandclassifysmoothmanifoldsaccordingtotheir cycles, or more generally submanifolds, that can be drawn without continuously deforming into eachother. Onecanviewthesecyclesascutswhichcanbegluedbacktogether,oraszippers which can be fastened and unfastened. Cycles are classified by dimension. For example, a linedrawnonasurfacerepresentsa1-cycle,whileasurfacecutthroughathree-dimensional manifoldisa2-cycle. GivenatopologicalspaceX,theboundaryoperatordefinesachaincomplexas ··· ∂ k+1 − −− →C k (X) ∂ k −−→C k− 1 (X) ∂ k− 1 − −− →··· ∂ 2 −−→C 1 (X) ∂ 1 −−→C 0 (X) ∂ 0 −−→ 0. The fact that the boundary of a boundary is empty implies thatim(∂ k+1 )⊆ ker(∂ k ), where im(·)andker(·)denotetheimageandthekernel,respectively. Thisleadstothedefinitionof thek-thhomologygroupas H k (X):=ker(∂ k )/im(∂ k+1 ). Theelementsofthekernelarecalledcycles. A chain complex is called exact if the image of the(k+1)-th map is always equal to the kernelofthek-thmap. Inthissense,thehomologygroupsofX providesameasurethathow closethechaincomplexassociatedtoX istobeingexact. WhenX isasimplicialcomplex,thedimensionofthek-thhomologycountsthenumber of“holes”inX atdimensionk. Onewayofcomputingitisthentoputmatrixrepresentations ofboundarymappingsintheSmithnormalform. Inagraph,forexample,H 1 (X)countsthe numberofloopsorcircuits. Cohomology groupsare formallysimilar to homologygroups, with thedifference that one startswithacochaincomplexas ··· d k+1 ← −− −C k (X) d k ←−−C k− 1 (X) d k− 1 ← −− −··· d 2 ←−−C 1 (X) d 1 ←−−C 0 (X) d 0 ←−− 0. 24 SmoothversusCombinatorialGeometry Since the coboundary maps satisfyd k+1 ◦ d k =0, we getim(d k )⊆ ker(d k+1 ), which leads to thedefinitionofthe k-thcohomologygroupas H k (X):=ker(d k+1 )/im(d k ). 2.2.6 Incidencematrixanddualcomplex Given ann-complexP, the k-incidence matrixB k (P) describes the correlation between all orientedk-cells and(k− 1)-cells inP. We define B k (P)∈R n k− 1 × n k so as to have one row for each (k− 1)-cell and one column for each k-cell. The componentB k (i,j) takes value 1 ifσ j k− 1 is a coherent boundary element forσ i k , it takes value− 1 ifσ j k− 1 is an incoherent boundary element forσ i k , and it is zero ifσ j k− 1 is not on the boundary ofσ j k . Observe that both the structure and the orientation of ann-complex is fully described by the collection of k-incidencematricesfor16k6n. Representing ak-chainc k as a column vector of sizen k , ak-cochainc k can be represented as a row vector of the same size, where the duality pairing betweenc k andc k is interpreted by a matrix product as c k (c k ) := ⟨c k ,c k ⟩ = c k c k . Then B k can be viewed as a linear transformationthatmapsak-chainintoitsorientedsetofboundaryelements. Thisisinfactthe algebraicrepresentationoftheboundaryoperator∂ k ,whichleadsto ∂ k =B k ∈R n k− 1 × n k . Similarly,theadjointofB k representsthealgebraiccoboundaryoperatoras d k =adj(B k )=B ⊤ k ∈R n k × n k− 1 . Givenaprimaln-complexP,wemaybuildadualn-complexP ∗ asfollows. Letusstart with a 0-cell inside eachn-cell at its centroid. The 1-cells of the dual complex are the edges connectingthe0-cellsinsidetwoadjacentn-cellsthatsharea(n− 1)-cellastheboundary. Then the 2-cells of the dual are the surfaces delimited by the 1-cells of the dual, the 3-cells of the dual are the volumes delimited by the 2-cells of the dual, and so on. Since each cell in the dual complexisidentifiedbyexactlyonecellintheprimalcomplex,theorientationoftheprimal cellscanbeusedtodefinetheorientationofthedualcells. By this definition of dual complex, for every k-cell of the primal complex there exists a corresponding(n− k)-cellofthedualcomplex. Inparticular,suchaconstructionpreservesthe 2.2CombinatorialGeometry 25 structureof theincidence matrix. Thus, thek-thincidence matrixon thedual complexsimply becomesthetransposeofthe(n− k+1)-thincidencematrixontheprimalcomplex, B k (P ∗ )=B ⊤ n− k+1 (P). Observe that this notation ofduality inheritsthe algebraicstructure ofthe primalcomplex. Further,thedualstructurebasesonlyontheconnectivitycondition,whichiscompletelyinde- pendent of the embedding. We remark that such a purely topological definition of dual complex embodiestheconceptofPoincaredualityoncohomologygroups. 2.2.7 CombinatorialHodgeduality OnaRiemannianmanifoldM,theHodge-staroperatorisalinearisomorphismfromk-formsto (n− k)-forms,represented by⋆:Ω k (M)→Ω n− k (M). Notethatby definition,both k-forms and(n− k)-formsareofequaldimensions. A similar isomorphism can be defined on an n-complex between each space ofk-chains ontheprimalmetrizedcomplex(P,g)andthespaceof(n− k)-chainsonthedualmetrized complex(P ∗ ,g ∗ ), leading to the combinatorial Hodge star operator⋆ :C k (P)→C n− k (P ∗ ). Recall that by the definition of dual complex, primal k-chains and dual (n− k)-chains are represented by vectors of the same dimension. Equivalently, the combinatorial⋆ operator maps k-cochainsonP into(n− k)-cochainsonP ∗ ,scaledbytheinversemetrictensorg k . Thatis, foranyk-cochainc k ∈C k (P) anditsdual(n− k)-cochainc k∗ ∈C n− k (P ∗ ), ⋆c k :=c k∗ =g k c k . Itiseasytoseethatthe⋆operationonadualcomplexistheinverseofthe⋆operationonits primalcomplex,uptoasign. Remark 2.3. ThecombinatorialHodgestarcombines thepurely topologicalinformationpro- videdbythedualcomplexwiththepurelygeometricalinformationprovidedbythemetrictensor. It is often desired to set up metric tensor on the dual complexg ∗ k such that the inner product betweendual(n− k)-cochainsisequivalenttotheinnerproductbetweenprimalk-cochains. Onecanverifythatsettingg ∗ k =g n− k satisfiessucharequirement. Thenadual (n− k)-cochain c k∗ ∈C n− k (P ∗ )correspondswithaprimalk-cochainc k ∈C k (P)throughc k =g k c k∗ ,which simplifiesthecombinatorial ⋆ operatortoadiagonalmatrix. 26 SmoothversusCombinatorialGeometry 2.2.8 Combinatorialcodifferentialoperator A very important application of the Hodge duality in smooth geometry is to define the cod- ifferential operator δ : Ω k (M) → Ω k− 1 (M), which is the adjoint of the exterior derivative d:Ω k (M)→Ω k+1 (M). Thatis,⟨η ,δξ ⟩=⟨dη ,ξ ⟩foranyξ ∈Ω k+1 (M)andη ∈Ω k (M). Notethatthecodifferentialsatisfies δ ◦ δ =0,inheritedfromd◦ d=0. Given a metrized complex(P,g), the combinatorial codifferential operator δ k :C k (P)→ C k− 1 (P) mapsak-cochainintoa(k− 1)-cochainandisdefinedby δ k :=⋆d n− k+1 ⋆, δ 0 (C 0 ):=0. Therightmost⋆operatormapsthegivenprimalk-cochainintoadual(n− k)-cochain. The d n− k+1 operator then maps the latter into a dual (n− k +1)-cochain. Finally, the second ⋆ operator maps the dual(n− k+1)-cochain into a primal(k− 1)-cochain. On a 3-complex, Fig. 2.1 illustrates how cochains live on the cells of primal and dual complexes, while the combinatorialoperatorsprovidethemappingbetweenthem. The duality equationB ⊤ n− k+1 (P ∗ ) = B k (P) on incidence matrices yieldsδ k = ⋆B k ⋆. When ⋆ operator is simply a diagonal matrix, the combinatorial codifferential operator can equivalentlybeexpressedbyann k− 1 × n k matrixas δ k =g ∗ n− k+1 B k g k =g k− 1 B k g k . It is easy to confirm that in the special case where the metric is Euclidean on the entire cell complex, the tensorg is simplified to the identity matrix and the codifferential matrix is simplifiedtothetransposeoftheexteriorderivativematrix. 2.2.9 CombinatorialLaplace-deRhamoperator The heart of the Hodge theory is Laplace-deRham operator∆ :Ω k (M)→Ω k (M) defined by ∆ :=(δ +d) 2 =δ ◦ d+d◦ δ ,whichissymmetric,⟨∆ ξ ,η ⟩=⟨ξ ,∆ η ⟩,andnonnegative, ⟨∆ η ,η ⟩ > 0, for any ξ ∈ Ω k+1 (M) and η ∈ Ω k (M). The Laplace-Beltrami operator ∆=div ◦ ∇=δ ◦ d is a special case of Laplace-deRham operator and the two operators turn tobeequivalent, uptoasign, whenactingonscalarfunctions. 2.2CombinatorialGeometry 27 ⋆ ⋆ ⋆ ⋆ ⋆ − 1 ⋆ − 1 ⋆ − 1 ⋆ − 1 Fig. 2.1Primalanddualcells,alongwiththecombinatorialmappingoperatorsona3-complex. Thereexistsadual3-cochain(polyhedron)foreachprimal0-cochain(vertex),adual2-cochain (polygon)foreachprimal1-cochain(edge),adual1-cochainforeachprimal2-cochain,andadual 0-cochainforeachprimal3-cochain. Thecoboundaryoperatordandthecodifferentialoperator δ mapbetweenk and(k+1)cochainsonthesamecomplexofeitherprimalordual,whilethe Hodge star operator⋆and its inverse⋆ − 1 swapcochainsbetweenprimalanddualcomplexes. Givenametrizedcomplex(P,g),wedefinethe combinatorialLaplace-deRhamoperator ∆ k :C k (P)→C k (P) inasimilar way, ∆ k :=δ k+1 ◦ d k+1 +d k ◦ δ k . Equivalentmatrixnotationisobtainedby ∆ k =g k B k+1 g k+1 B ⊤ k+1 +B ⊤ k g k− 1 B k g k ∈R n k × n k . 2.2.10 Weightedcomplex Analogous to the weighted smooth manifolds, one may attribute a weight to eachk-cell in a cell complex. Define a k-weight function w k :P k → R + that operates on a k-cellσ k ∈P k and assigns it with a positive scalarw k (σ k ). Compose a diagonal matrixw k =diag(w k (σ k )), wherethei-thdiagonalentryequalstheweight ofthei-thk-cell,and letw P bethecollection ofallsuchmatricesonthecellcomplexP. Aweightedmetrizedcomplexisthendefinedbya triple(P,g P ,w P ) withg P beingthecollectionofallk-metricsonP. 28 SmoothversusCombinatorialGeometry As expected, the combinatorial exterior derivative d k remains the same on a weighted complex. However,theweightedcodifferential operatoron(P,g P ,w P ) ismodifiedas δ k w :=g k− 1 w k− 1 B k w − 1 k g k . Similarly,thecombinatorial weightedLaplace-deRhamoperatorisdefinedby ∆ k w :=g k w k B k+1 w − 1 k+1 g k+1 B ⊤ k+1 +B ⊤ k g k− 1 w k− 1 B k w − 1 k g k . 2.3 Graphas1-complex Consider a connected, weighted graph with set of nodesV and set of undirected edgesE, which representsa1-complex. Notingthatδ 0 w =0bydefinition,weobtaintheweightedgraphLaplace- deRham operator ∆ 0 w = g 0 w 0 B 1 w − 1 1 g 1 B ⊤ 1 . Taking Euclidean metric on the graph, the tensorsg 0 andg 1 are simplified to the identity matrices, which leads to ∆ 0 w =w 0 B 1 w − 1 1 B ⊤ 1 . Further assuming that all nodes are equally weighted by 1, we get∆ 0 w =B 1 w − 1 1 B ⊤ 1 , which is knownas graphLaplacian andisoftendenotedbyL. Letusdropsubscripts1forbrevityanddenoteL=Bw − 1 B ⊤ . Onecanverifythat x ⊤ Lx= X ij∈E w − 1 ij (x i − x j ) 2 >0 which meansL is positive semi-definite. ObservingL1 =0 implies that1 is an eigenvector corresponding to the smallest eigenvalueλ 1 = 0. For the second smallest eigenvalueλ 2 , let ν be the eigenvector orthogonal to1, meaning thatλ 2 =ν ⊤ Lν = P ij∈E w − 1 ij (ν i − ν j ) 2 and ν ⊤ 1=0. AsL is positive semi-definite, λ 2 > 0. Let us assume λ 2 = 0. Since the graph is connected, there exists a path between every two nodes. Considering this along with the assumption ofλ 2 =0 entailν =c1, wherec is a scalar. The latter contradicts the requirement ofν ⊤ 1=0,whichimpliesλ 2 >0. LetL † betheMoore-PenrosepseudoinverseofL. BothLandL † havethesameeigenvectors, whiletwocorrespondingeigenvaluesarereciprocalsofeachother,exceptthat1replacesthe zeroeigenvalueofL. Onecanverifythat L † = L+ 1 n 1 ⊤ 1 − 1 − 1 n 1 ⊤ 1 with n:=|V|. Further,L † enjoysthestructuralpropertyofL † 1=0. 2.3Graphas1-complex 29 The Dirichlet LaplacianL ◦ :=B ◦ diag(σ )B ◦ ⊤ is made fromL by discarding the entries correspondingtoanarbitraryreferencenoded. LetE =E 1 +E ◦ ,whereE ◦ isthesetofedges withoneendconnectedtonoded. Onecanverifythat∀x̸=0, x ⊤ L ◦ x= P ij∈E 1 σ ij (x i − x j ) 2 + P id∈E◦ σ id x 2 i >0 whichmeansL ◦ ispositivedefinite andsoinvertible. ThisalsoimpliesthatB ◦ isafullrowrank matrix. Further,onecanverifythatL ◦ 1=y isanonzerovectorwithnon-negativecoordinates, wherey i =0 ifid∈E 1 andy i >0 ifid∈E ◦ . Remark 2.4. It is sometimes misunderstood thatL ◦ carries the same eigenvalues asL but the zero,whichisnottrue. ByKirchhoff’stheorem,thenumberofspanningtreesisequaltoanycofactorof L. Since every connected graph has at least one spanning tree, deleting any row and any column fromL results in a matrix of nonzero determinant, and soL ◦ is full rank. This gives another way to confirmthat L ◦ ispositivedefiniteforanyconnectedgraph. 2.3.1 Resistivenetworkinterpretation Consideranelectricnetworkassociatedtoaconnectedweightedgraph,whereeachedgeij∈E isreplacedbyalinearlumpedresistorwiththeconductanceofσ ij . Letelectricalcurrentsbe injectedinto(resp. drawnfrom)thenetworkbypositive(resp. negative)independentcurrent sources attached to the nodes. Let a noded be grounded at zero voltage potential and let no other grounded node exist in the network. (Note the difference between a grounded node and a node with zerovoltage.) At the nodes otherthand, letu ◦ be the vector of independent current sourcesandv ◦ thevectorofinducedvoltages. R1: CombiningKirchhoff’sCurrentLaw(KCL)andOhm’slaw,weobtain u ◦ =L ◦ v ◦ and v ◦ =L − 1 ◦ u ◦ . R2: One may envision d as a node, also referred to as sink, that collects the net current injected into the network, i.e., algebraic sum of current sources, and feeds it back to the sources soastomakeanadiabaticprocess. Thisleadsto u=Lv with u d =− P i∈V\{d} u i and v d =0. 30 SmoothversusCombinatorialGeometry LetL † betheMoore-PenrosepseudoinverseofL. Onecanverifytheinverserelationshipas v =L † u− c1 with c:=(L † u) d =− P i∈V\{d} (L † u) i . R3: Passivity: Foru ◦ ̸= 0, the cumulative energy spent by all current sources is always positive,i.e.,u ◦ ⊤ v ◦ >0. Thisdoesnotimplythattheenergyspentbyeachindividualcurrent sourceisalwayspositivetoo. R4: Ifu ◦ <0(resp.u ◦ 40),thenv ◦ <0(resp.v ◦ 40). Further,v ◦ ≻ 0(resp.v ◦ ≺ 0),if currentsarenonzeroatthenodesneighbortoground. Theconverseisnottrue,i.e.,imposing positive (resp. negative) voltages at all nodes could entail negative (resp. positive) current sourcesat somenodes. R5: Eachpositive(resp. negative)currentsourcecreatesthelargestpositive(resp. negative) voltage at the node to which it is connected. The magnitude of created voltage at the nodes reducealongeverypossiblepathtowardsthegroundinproportiontotheedgeconductances. R6: Superposition of sources: The voltage at each node equals the algebraic sum of the voltagescreatedbyeachcurrentsourceseparatelywhileallothersourcesarezero. R7: Superposition ofresistors: Given aresistive networkG, considertwo sub-networksG 1 andG 2 withthesamesetofnodes,edgesandcurrentsources,butwiththeresistanceofeach edgeinG beingequaltotheparallelofcorrespondingresistancesinG 1 andG 2 . Thevoltageat eachnodeinG equalsthealgebraicsumofthecorrespondingvoltagesinG 1 andG 2 . Thecurrent througheachedgeinG equalsthealgebraicsumofthecorrespondingcurrentsinG 1 andG 2 . R8: The effective electrical resistance between two nodes i andj, using the Moore-Penrose pseudoinverse,isgivenby r eff ij =L † ii +L † jj − 2L † ij . Alternatively,theDirichletLaplacianprovides r eff ij =    (L − 1 ◦ ) ii +(L − 1 ◦ ) jj − 2(L − 1 ◦ ) ij ifi,j̸=d (L − 1 ◦ ) ii otherwise. Rayleigh’s monotonicity principle states that if the resistance of one edge increases (resp. decreases), the effective resistance between every two nodes also increases (resp. decreases) or remainsunchanged. Chapter3 Heat-Diffusion: ParetoOptimalDynamic Routing Adynamicroutingpolicy,referredtoasHeat-Diffusion(HD),isdevelopedformultihopuniclass wirelessnetworkssubjecttorandomtraffic,time-varyingtopologyandinter-channelinterference. The policy uses only current condition of queue occupancies and channel states, with requiring no knowledge of traffic and topology. Besides throughput optimality, HD minimizes an average quadraticroutingcostdefinedbyendowingeachchannelwithatime-varyingcostfactor. Further, HDminimizesaveragenetworkdelayintheclassofroutingpoliciesthatbasedecisionsonlyon currentconditionoftrafficcongestionandchannelstates. Further,inthisclassofroutingpolicies, HD provides a Pareto optimal tradeoff between average routing cost and average network delay, meaning that no policy can improve either one without detriment to the other. Finally, HD fluid limitfollowsgraphcombinatorialheatequation,whichcanopenanewwaytostudywireless networksusingheatcalculus,averyactiveareaofpuremathematics. 3.1 Introduction Throughput optimality, which means utilizing the full capacity of a wireless network, is critical torespondtoincreasingdemandforwirelessapplications. Theseminalworkin[44]showed that the link queue-differential, channel rate-based Back-Pressure (BP) algorithm is throughput- optimalunderverygeneralconditionsonarrivalratesandchannelstateprobabilities. Follow-up worksshowed thattheclassof throughput-optimalroutingpoliciesisindeedlarge[45–48]. The 32 Heat-Diffusion: ParetoOptimalDynamicRouting challengeisthentodeveloponethat,inaddition,isoptimalrelativetosomeotherimportant routingobjectives. WeproposeHeat-Diffusion(HD), a throughput-optimalroutingpolicythatoperates under the same general conditions and with the same algorithmic structure, complexity and overhead asBP,whilealsoholdingthefollowingimportantqualities: • HD minimizestheaverage quadraticroutingcostR inthesenseof Dirichlet. Endowing eachwirelesslinkwithatime-varyingcostfactor,wedefinethe Dirichletroutingcostasthe productofthelinkcostfactorsandthesquareoftheaveragelinkflowrates. Suchageneric routingcostmayreflectdifferenttopology-basedpenalties,e.g.,channelquality,routing distance and power usage, even a cost associated with greedy hyperbolic embedding [49]. • HDminimizesaveragetotalqueuecongestionQ,whichisproportionaltoaveragenetwork delay by Little’s Theorem, within the class of routing algorithms that use only current queueoccupanciesandcurrentchannelstates,possiblytogetherwiththeknowledgeof arrival/channelprobabilities. • Inthesameclass,HDoperatesontheParetoboundaryofperformanceregionbuilton theaverage network delayQ andtheaveragequadraticroutingcostR andcanbemade to movealong this boundary by changinga control parameterβ that compromises between thetwoobjectivesQ andR (seeFig.3.1). 3.1.1 Relatedworks ThestudyofBPschemeshasbeenaveryactiveresearchareawithwide-rangingapplicationsand manyrecenttheoreticalresults. Inpacketswitches,congestion-basedscheduling[46,50,51]was extendedtoadmitmoregeneralfunctionsofqueuelengthswithparticularinterestonα -weighted schedulersusingα -exponentofqueuelengths[45]. Asanotherextensioninpacketswitches, [47]introducedProjectiveConeSchedulers(PCS)toallowschedulingwithnondiagonalweight assignments. The work in [48] generalized PCS using a tailored “patch-work” of localized piecewisequadraticLyapunovfunctions. In wireless networks, shadow queues enabled BP to handle multicast sessions with reduced numberofactualqueuesthatneedtobemaintained[52]. Replacingqueue-lengthbypacket-age, [53] introduced a delay-based BP policy. To improve BP delay performance, [54] proposed place-holders with Last-In-First-Out (LIFO) forwarding. Adaptive redundancy was used in [55] 3.1Introduction 33 toreducelighttrafficdelayinintermittentlyconnectedmobilenetworks. Usinggraphembed- ding,[49]combinedBPwithagreedyroutingalgorithminhyperboliccoordinatestoobtaina throughput-delaytradeoff. There have been several reductions of BP to practice in the form of distributed wireless protocols of pragmatically implemented and experimentally evaluated [56–58]. Some attempts havealsobeenmadetoadopttheBPframeworkforhandlingfinitequeuebuffers[59]. SimilartoBP,alsoHDrestsonacentralizedschedulingwithacomputationalcomplexity that canbe prohibitivein practice. Fortunately,muchprogresshasrecentlybeenmadetoease thisdifficultybyderivingdecentralizedschedulerswiththeperformanceofarbitrarilyclose to thecentralizedversionasafunctionofcomplexity[60–62]. 3.1.2 Contributions We derive HD from combinatorial analogue of classical heat equation on smooth manifolds, whichleadstothefollowingkeycontributions: (Fluid)Translating“queueoccupancymeasuredinpackets”to“heatquantitymeasuredin calories,” the fluidlimit ofinterference HDflow mimicsasuitably-weightednon-interference heatflow,inagreementwiththesecondprincipleofthermodynamics. Indoingso,weintroducea newparadigmthatmightbecalled“wirelessnetworkthermodynamics,”whichbuildsarigorous connection between wireless networking and well-studied domains of physics and mathematics. (Cost) HD reduces the Dirichlet routing cost to its minimum feasible value among all stabilizing routing algorithms. To the best of our knowledge, this is the first time a feasible routing algorithm asserts the strict minimization of a cost function subject to network stability, i.e.,boundedqueueoccupanciesandnetworkdelay. Thisiswhilethedrift-plus-penaltyapproach of [63], as the best-known alternative, can get only close to the minimum of this routing cost at theexpenseofinfinitelylargenetworkdelay. (Delay) HD minimizes average queue lengths, and so average network delay, within the classofroutingalgorithmsthatactbasedonlyoncurrentconditionofqueueoccupanciesand channel states, including those with the perfect knowledge of arrival/channel probabilities. This important class contains stationary randomized algorithms [63], original BP policy [44] and mostBPderivations[45–50,52–59,61,62,64]. (Pareto) In the class of algorithms defined in (Delay), let the performance region built on average delay and the Dirichlet routing cost be convex. Then HD operates on the Pareto boundaryofthisregionwhiletheoptimaltradeoffcansolelybecontrolledbyaroutingparameter 34 Heat-Diffusion: ParetoOptimalDynamicRouting independently of network topology and traffic. This means that no other policy in this class can makeabettercompromisebetweenthesetworoutingobjectivesandthatanydeviationfrom HDoperationleadstothedegradationofatleastoneofthem. (Complexity) Last but not least, HD enjoys the same algorithmic structure, complexity and overhead as BP, giving it the same wide-reaching impact. This also provides an easy way to leveragealladvancedimprovementsofBPtofurtherenhanceHDquality. Atthesametime, it simplifiesthewaytopracticeviaasmoothsoftwaretransitionfromBPtoHD. 3.1.3 Organization Nextsectionprovidespreliminariesandproblemdefinitions. HDroutingpolicyisintroduced inSec.3.3followedbysomeillustrativeexamples. Section3.4presentsHDkeyproperty—a foundationtoallotherHDproperties. UsingLyapunovstability,HDthroughputoptimalityis proveninSec.3.5. WeshowinSec.3.6thatHDminimizesaveragenetworkdelayinanimportant classofroutingpolicies. Physics-orientedmodelofheatprocessondirectedgraphsisproposed in Sec. 3.7. Using fluid limit theory, Sec. 3.8 shows that in limit, HD packet flow resembles combinatorial heat flow on its underlyingdirectedgraph. Using heatcalculus, Sec.3.9 shows thatHDstrictlyminimizestheDirichletroutingcost. ParetooptimalityofHDperformanceis discussedinSec.3.10. ThechapterisconcludedinSec.3.11. 3.1.4 Notation Wedenotevectorsbyboldlowercaseandmatricesbyboldcapitalletters. By0wedenotethe vector of all zeros, by1 the vector of all ones, and byI the identity matrix. On arrays: min and maxaretakenentrywise;4and<expressentrywisecomparisons;and⊙ denotestheSchur product. Forv as a vector,v ⊤ denotes its transpose, diag(v) its diagonal matrix expansion, ∥v∥itsEuclideannorm,andv + :=max{0,v}. ForS asaset,|S| denotesitscardinality. We useI as the scalar indicator function, andII v≻ 0 as the vector indicator function that its entry i takes the value 1 if v i >0, and 0 otherwise. By ˙ x(t) we denote the time derivative of x(t). For a variablex related toa directededgeℓ from nodei to nodej, we use notationsx ℓ andx ij interchangeably. We usein(i) andout(i) to denote the sets of nodes neighbor to nodei with respectivelyincominglinkstoandoutgoinglinksfromnodei. With⌊x⌋and⌈x⌉werespectively denotethe floorandtheceilingofarealnumber x. 3.2Preliminaries 35 3.2 Preliminaries In this chapter, we consider a uniclass wireless network that operates in slotted time with normalized slotsn∈{0,1,2,···} . The network is described by a simple, directed connectivity graph with set of nodesV and directed edgesE. New packets, all with the same destination atnoded∈V,randomlyarriveintodifferentnodes,requiringamultihoproutingtoreachthe destination. Wireless channels may change due to node mobility or surrounding conditions. AssumingthesetsV andE changemuchslowerthanchannelstates,wefixthemduringthetime ofourinterest;atemporarilyunavailablelink(dueto,e.g.,obstacleeffectandchannelfading)is then characterized by zero link capacity. Extended mobility that can lead to permanent change in network topology is not considered here. We assume that channel states remain fixed during atimeslot,whiletheymaychangeacrossslots. In wireless networks, transmission over a channel can happen only if certain constraints areimposedontransmissionsovertheotherchannels. Aninterferencemodelspecifiesthese restrictions on simultaneous transmissions. We consider a family of interference models under which a node cannot transmit to more than one neighbor at the same time. Thus, in a most generalcase,anodemayreceivepacketsfromseveralneighborswhilesendingpacketsoverone of its outgoing links. Interference constraints used by all well-known network and link layer protocols,includinggeneralK-hopinterferencemodels,fallinthisfamily. Definition 3.1. Given an interference model, a maximal schedule is such a set of wireless channelsthatnotwochannelsinterferewitheachotherandnomorechannelcanbeaddedto thesetwithoutviolatingthemodelconstraints. We describe a maximal schedule with a scheduling vectorπ ∈{0,1} |E where π ij = 1 if channelij isincluded,andπ ij =0 otherwise. Definition 3.2. Given a connectivity graph (V,E), scheduling set Π is the collection of all maximalschedulingvectors. Definition 3.3. WithE denoting expectation, the expected time average of a discrete-time stochasticprocessx(n) isdefinedas x:=limsup τ →∞ 1/τ X τ − 1 n=0 E{x(n)}. (3.1) Definition 3.4. A queuing network is stable if queue at each nodei and at each slotn, denoted asq i (n),hasaboundedtimeaverageexpectation,viz.,q i <∞. 36 Heat-Diffusion: ParetoOptimalDynamicRouting Definition 3.5. Given a wireless network, an arrival vectora(n) is stabilizable if there exists a routingpolicythatcanmakethenetworkstableundera(n). Foralinkij,itscapacityµ ij (n),whichisfrequentlycalledtransmissionrateinliterature, countsthemaximumnumberofpacketsthelinkcantransmitatslotn. Thelinkactualtrans- missionf ij (n), on the other hand, counts the number of packets genuinely sent over the link at slotn. Each link is also endowed with a cost factorρ ij (n)> 1 that represents the cost of transmitting one packet over the link at slotn; for example,ρ ij =ETX ij , withETX as defined in[65],oracostassociatedwithgreedyembedding[49]. 3.2.1 Problemstatement Foraconstraineduniclassnetworkdescribedabove,weproposeHDroutingpolicythatsolvesthe threestochasticoptimizationproblemsasfollows. Itisimportanttonotethattheseoptimization problemsmustbesolvedatnetworklayeralone,whichmakesittotallydifferentfromcross-layer optimization[66–69]thataimstocontrolcongestionbycontrollingarrivalratesintonetwork layer. With no control on arrivals, the basic assumption here is that arrival rates lie within networkcapacityregion,makingtheroutingsystemstabilizable. Obviously,nothingprevents one to either install a flow controller on top of HD or develop an HD-based Network Utility Maximization(NUM)protocol. (Delay) Averagenetworkdelayminimization: Minimize Q:= X i∈V q i . (3.2) SolvingthisproblemforageneralcaserequirestheMarkovstructureofnetworktopology process, plus arrival and channel state probabilities. Then in theory, the solution is obtained through dynamic programming for each possible topology along with solving a Markovdecisionproblem. Byevenhavingallthisrequiredinformation,thenumberof queue backlogs and channel states increase exponentially with the size of network, which makes dynamic programming and Markov decision theory prohibitive in practice. In fact, even for the case of a single channel, it is hard to implement the resulting stochastic algorithms[70]. Whilehavingapracticalsolutionforageneralcaseseemsdubious,we show in Th. 3.3 that HD policy solvesthisproblemwithinanimportantclassofrouting algorithms,withoutrequiringanyoftheabove-mentionedinformationordealingwith anydynamicprogramingorMarkovdecisionprocess. 3.2Preliminaries 37 (Cost) Averagequadraticroutingcostminimization: Minimize R := X ij∈E ρ ij f ij 2 . (3.3) ThelossfunctionR,byconcept,spreadsouttrafficwithaweightedbiastowardslower penaltylinksthatremindstheoptimaldiffusionprocessesinphysics,suchasheatflowand electrical current [71]. It is shown in [72,63] that a stationary randomized algorithm can solvethisproblem. Whilesuchan algorithm existsin theory,it isintractablein practice asitrequires a fullknowledgeoftrafficandchannelstateprobabilities. Further, assuming all of the probabilities could be accurately estimated, the network controller still needs to solve a dynamic programming for each topology state, where the number of states grows exponentially with the number of channels. Nonetheless, we show in Th. 3.8 that HD policysolvesthisproblemwithoutrequiringanyknowledgeoftrafficandchannelstate probabilitiesordealingwithanydynamicprogramming. (Pareto) Paretooptimalperformance: Minimize (1− β )Q+β R (3.4) whereβ ∈[0,1] is a control parameter to determine relative importance between average delay and average routing cost, which naturally plays the role of Lagrange multiplier too. Toourknowledge,thisisthefirsttimeinliteraturethatsuchamulti-objectiveoptimization problemisaddressedinthelevelofsolelynetworklayer. Whileeventherelatedsingle- objectiveoptimizationproblemsarenoteasytomanage,weshowinTh.3.9thatwithinthe sameclassofroutingpoliciesmentionedin(Cost),HDpolicysolvesproblem(3.4)subject to convex Pareto boundary on the feasible(Q,R) region, with requiring no knowledge of trafficandtopology. 3.2.2 UniclassBack-Pressure(BP)policy Ateachslotn, originaluniclassBP [44] observes queue backlogsq i (n) at network layerand estimateschannelcapacitiesµ ij (n) tomakearoutingdecisionasfollows. 38 Heat-Diffusion: ParetoOptimalDynamicRouting 1) UniclassBPweighting: Foreverylinkij findthelinkqueue-differential q ij (n):=q i (n)− q j (n) andweightthelinkwith w ij (n):=µ ij (n)q ij (n) + . 2) UniclassBPscheduling: Findaschedulingvectorsuchthat π (n)=argmax π ∈Π X ij∈E π ij w ij (n) wheretiesarebrokenarbitrarily. 3) UniclassBPforwarding: Overeach activatedlinkwithw ij (n)>0transmitpackets atfull capacityµ ij (n). Ifthereisnoenoughpacketsatnodei,transmitnullpackets. 3.2.3 UniclassV-parameterBPpolicy Thusfar,thedrift-plus-penaltyapproach[72,63],whichwerefertoasV-parameterBPhereafter, has been the only feasible approach to decreasing (not minimizing) a generic routing penalty at networklayer. WetaketheV-parameterBPasayardstickastohowHDperforms. Toincorporate averageroutingcostR intotheoriginalBP,theV-parameterBPaddsausagecosttoeachlink queue-differentialviareplacingthelinkweightofBPby w ij (n):=µ ij (n) q ij (n)− Vρ ij (n)µ ij (n) + (3.5) whereV ∈ [0,∞) makes a compromise on queue occupancy in the favor of routing penalty, withV =0 recoveringtheoriginalBPwithnoroutingpenaltyconsideration. The V-parameter BP yields a Dirichlet routing cost withinO(1/V) of its minimum feasible value to the detriment of growing average delay ofO(V) relative to that of original BP [63]. Thus, the policy is not able to achieve minimumrouting cost subject to finite network delay, i.e., delay grows to infinity as routing cost is pushed towards its minimum. Another issue is that the resultingtradeoffdependsonboth V andthenetwork,withtwonegativeconsequences: – ThesamevalueofV leadstodifferentlevelsoftradeoffindifferentnetworks. – The level of tradeoff in the same network varies by topologyand arrival rates, making it difficulttofindaproper V inpractice. 3.3UniclassHeat-Diffusion(HD)Policy 39 3.3 UniclassHeat-Diffusion (HD) Policy Toprovideaconvenientwayofunifyingourproposedschemewiththelargebodyofprevious worksonBP,wedesignHDwiththesamealgorithmicstructure,complexityandoverhead,in bothcomputationandimplementation,asBP. 3.3.1 UniclassParetooptimalHDalgorithm At each slotn, uniclass HD policy observes link queue-differentials q ij (n):=q i (n)− q j (n) at network layer and estimates channel capacitiesµ ij (n) and channel cost factorsρ ij (n) to make a routingdecisionasfollows. 1) Uniclass HD weighting: For every link ij first calculate the number of packets it would transmitifitwereactivated: ̂ f ij (n):=min φ ij (n)q ij (n) + , µ ij (n) φ ij (n):=(1− β )+β/ρ ij (n) (3.6) where the Lagrange control parameterβ is as defined in (3.4) to make a tradeoff between queueoccupancy androuting penalty, andthe hatnotationdenotes apredicted valuewhich wouldnotnecessarilyberealized. Thendeterminethelinkweightas w ij (n):=2φ ij (n)q ij (n) ̂ f ij (n)− ̂ f ij (n) 2 . (3.7) 2) UniclassHDscheduling: Findaschedulingvector,inthesamewayasBP,usingthemax- weightscheduling,suchthat π (n)=argmax π ∈Π X ij∈E π ij w ij (n) (3.8) wheretiesarebrokenarbitrarily. 3) UniclassHDforwarding: Over eachactivatedlinktransmit ̂ f ij (n) numberofpacketsas f ij (n)=    ̂ f ij (n) if π ij (n)=1 0 otherwise (3.9) wheref ij (n) representsthenumberofpacketsgenuinelysentoverlinkij atslotn. 40 Heat-Diffusion: ParetoOptimalDynamicRouting Itiscriticaltodiscriminateamongactuallinktransmissionsf ij (n),linktransmissionpre- dictions ̂ f ij (n) and link capacitiesµ ij (n). Also notice that ̂ f ij (n) in (3.9) could be non-integer. In practice, the number of packets to be transmitted over links can be rounded to the nearest integerwith noimportantinfluence onthe performance. To be more precise, however, every node may algebraically add the packet residuals sent on each of its ongoing links so as to make acompensationassoonasthesumhitseither1 or− 1. Table1comparesHDandV-parameterBPalgorithms,whichemphasizesthesamealgorith- micstructure,computationalcomplexityandoverheadsignaling. Remark 3.1. Some general notes: (i) Sinceρ ij (n)>1 by assumption, we get0<φ ij (n)61 forallβ ∈[0,1]. (ii)Ifq ij (n)60,weget ̂ f ij (n)=0 dueto(3.6)andw ij (n)=0 from (3.7);in thiscase,evenifthelinkwerescheduledby (3.8),stillnopacketwouldbetransmittedoverit. (iii) Ifq ij (n)>0, we getq ij (n) + =q ij (n) and since ̂ f ij (n)6φ ij (n)q ij (n) due to (3.6), the link weight (3.7)stillremainspositive. (iv)Inlightofq ij (n) + 6q i (n)andφ ij (n)61,thevalueof ̂ f ij (n) neverexceedsthenumberofpacketsinthetransmittingnodei. Remark 3.2. Inaspecialcasethatalllinksareofthesamecapacity,i.e.,µ ij (n)=µ (n),and all linkqueue-differentials remainless than it, i.e., q ij (n)< µ (n), HD policy withβ = 0 and α -weighted policyof [45]withα =2 turn tobe equivalent. Packet switchesare wellsuited to thisspecialcase. Itwassuggestedin[45]thatasmallerα mayleadtoalowernetworkdelay, withanon-provenconjecturethatheavytrafficdelayisminimizedwhen α →0. Adiscussion ofthiswasgivenin[64]alongwithsomecounterexamples. Eveniftheconjectureweretrue, notethatforamultihoproutingproblem,therequirementofq ij (n)<µ ij (n)wouldimplythe networknottobeinaheavytrafficcondition. 3.3.2 HighlightsofParetooptimalHDdesign H1: While BP is derived by link capacity µ ij (n), HD emphasizes on actual number of transmittable packets ̂ f ij (n), though it also implicitly takes the link capacity into account through (3.6). Thus, HD allocates resources based only on genuinely transmittable packets, withoutcountingonnullpacketsasbeingpracticedinBPschemes. H2: Thelinkweight(3.7),whichitselfdirectlycontrolstheschedulingoptimizationproblem, is taken quadratic in the link queue-differential q ij (n), where for φ ij (n)q ij (n) 6 µ ij (n) is simplified to w ij (n)=φ ij (n) 2 q ij (n) 2 . This contrasts with BP weightingw ij (n)=µ ij (n)q ij (n) which is linear inq ij (n). The quadratic weight is central to HD key property (Th. 3.1) which is fundamentaltootherHDqualities. 3.3UniclassHeat-Diffusion(HD)Policy 41 Table 3.1 Algorithmic structureofuniclassParetooptimalHDversusV-parameterBP. Weighting ̂ f ij (n) V-BP min µ ij (n), q i (n) HD min 1− β +β/ρ ij (n) q ij (n) + , µ ij (n) w ij (n) V-BP µ ij (n) q ij (n)− Vρ ij (n)µ ij (n) + HD 2 1− β +β/ρ ij (n) q ij (n) ̂ f ij (n)− ̂ f ij (n) 2 Scheduling π (n)=argmax π ∈Π X ij∈E π ij w ij (n) Forwarding f ij (n)= ( ̂ f ij (n) if π ij (n)=1 0 otherwise H3: Varyingthepenaltyfactorβ makesauniversaltradeoffinperformancethatdepends neitheronnetworknoronarrivalswiththefollowingsignificantresults: • HDisthroughput-optimalforallβ ∈[0,1] (Th.3.2). • At β = 0, the average total queue Q, and so average network delay, decrease to their minimum feasible values within the class of routing policies that rely only on present queuebacklogsandcurrentchannelstates(Th.3.3). • Raisingβ adds to average delay in return for a lower routing cost, where the exclusive meritofHDistoprovidethebesttradeoffbetweenthesetwocriteria(Th.3.9). • Atβ =1,theaverageroutingcostR reachesitsminimum(Th.3.8)throughanoptimal tradeoffwithaveragenetworkdelay. Notethat inV-parameterBP,network delay grows toinfinityasroutingcostispushedtowardsitsminimum. H4: UnlikeBPthatforwardsthehighestpossiblenumberofpacketsoveractivatedlinks, HDcontrolspacketforwardingbylimitingittoφ ij (n)q ij (n)withmaximumφ ij =1atβ =0 and minimumφ ij =1/ρ ij atβ =1. This reduces queue oscillations by decreasing unnecessary packetforwardingacrosslinks,whichitselfreducestotalpowerconsumptionandroutingpenalty. Thus, it is not surprising to see that φ ij is decreasing, and so as to have a higher impact, by increasingβ that means more emphasis on routing penalty. Forwarding a portion of link queue- differentials rather than filling up link capacities also complies with resembling heat flow on the underlying directed graph (Th. 3.5) that in effect minimizes time average routing cost in light of Dirichlet’sprinciple(Th.3.8). Figure3.1providesagraphicalcomparisonbetweenoperationofHDforβ ∈[0,1]andV- parameterBPforV ∈[0,∞). TheperformanceregionisrestrictedtothesetofallQachievable 42 Heat-Diffusion: ParetoOptimalDynamicRouting 0   1    Average network delay Feasible space of objectives Average routing cost V = 0 (Original BP)  V Q R Fig.3.1GraphicaldescriptionofHDParetooptimalitywithrespecttoaveragequeuecongestion andtheDirichlet routing cost, comparedwiththeperformanceofV-parameterBP. bytheclassofallroutingpoliciesthat act based only on present queue backlogs and current channelstates,andisassumedtohaveaconvexParetoboundary. 3.3.3 Illustrativeexamples Inordertofocusmerelyonthepolicyitself,wetakeeverythingdeterministicinourexamples here, resting assure that the results purely show the policy performance not contaminated by stochasticeffects. WehoweverknowthatallHDpropertiesareanalyticallyprovenforstochastic arrivalsandrandomtopologiesunder verygeneralconditions. Two-queue downlink: Consider a base station that transmits data to two downlink users, where at most one link can be activated at each timeslot. Let link 1 be of constant capacity µ 1 =3 (packets/slot) and link 2 of time-varying capacityµ 2 >2. Assume one packet to arrive for each user at every timeslot. It is then easy to verify that forµ 2 <1.5, the given arrival goes beyondthenetworkcapacityregion. Forq 1 (0)=q 2 (0)=0,Fig.3.2comparestheperformanceofHDwithβ =0andoriginalBP. Theleftsidepaneldepictstimeslotevolutionofq 1 (n)+q 2 (n)forµ 2 =18. Therightsidepanel showsthesteady-stateaverageoftotalqueuelengthasafunctionofµ 2 . For26µ 2 65,both HDandBPperformthesame. Forµ 2 >5,however,averagetotalqueuelengthincreaseslinearly inµ 2 underBP,whileHDholdstheoptimalperformanceforalladmissiblelinkcapacities. This exemplifies H1 in the previous subsection, i.e., the efficiency of link scheduling based on actual transmittablepacketsratherthanlink capacities. Lossy link network: Consider the 4-node network of Fig. 3.3 with lossy links and subject to 1-hopinterferencemodel,i.e.,twolinkswithacommonnodecannotbeactivatedatthesame 3.3UniclassHeat-Diffusion(HD)Policy 43 0 5 10 15 20 0 1 2 3 4 5 6 7 0 10 20 30 0 1 2 3 4 5 6 7 BP HD (packets/slot) 2  Arrival beyond capacity region Average total queue length Total queue length timeslot BP HD BP HD Fig.3.2Two-queuedownlink: PerformanceofHDwithβ = 0versusoriginalBP.Whileforall admissible link capacities total queueisminimizedunderHD,itgrowslinearlyinµ 2 underBP. time. The links are labeled with both ETX and capacity, where ETX is a quality metric defined as the expected number of data transmissions required to send a packet without error over a link[65]. Assumethatateverytimeslotasinglepacketarrivesatnode1destinedfornoded. Following[56],letustakeρ ij =ETX ij . Forzeroinitialconditions,Fig.3.3comparestheperformanceofParetooptimalHDwith V-parameter BP. While HD easily stabilizes total queued packets at 1 for anyβ > 0, trying with different values of V indicates the weakness of V-parameter BP in aptly supporting the arrival. Thissimplisticallyshowsoneoftheimpactsofenteringlinkcostfactorρ ij asamultiplicand 0 5 10 15 20 25 30 0 5 10 15 20 25 data1 data2 data3 data4 Total queue length timeslot ETX=5 μ=5 μ=2 ETX=1 μ=2 ETX=1 μ=2 ETX=1 μ=2 ETX=1 1 BP (V=2) HD ( β>0) BP (V=1) BP (V=3) 0 5 10 15 20 25 30 0 5 10 15 20 25 data1 data2 data3 data4 0 5 10 15 20 25 30 0 5 10 15 20 25 data1 data2 data3 data4 0 5 10 15 20 25 30 0 5 10 15 20 25 data1 data2 data3 data4 0 5 10 15 20 25 30 0 5 10 15 20 25 data1 data2 data3 data4 d Fig.3.3Lossylinknetwork: PerformanceofParetooptimalHDversusV-parameterBP.While totalqueued packets is stabilized at1underHDforanyβ > 0,itgrowslinearlyinV underBP. 44 Heat-Diffusion: ParetoOptimalDynamicRouting in the HD weighting formula (3.7) rather than an addend in the V-parameter BP weighting formula (3.5). Power minimization: Consider the sensor network of Fig. 3.4 subject to 1-hop interference model. Suppose that each linkij has a noise intensityN ij ∈[1,5] which is randomly assigned at first and keeps fixed during the simulation. For each link, we adopt Shannon capacity µ ij = Ω ij log 2 (1+P ij /N ij ) withP ij as power transmission andΩ ij as bandwidth. At every timeslot, two packets arrive at nodes1, 2, 3 and 4, destined for noded. The aim is to minimize totalρ ij (f ij ) 2 withρ ij :=P ij /µ ij ,whichimplicitlyminimizestotalpowerconsumptioninthe network. Forsimplicity,letusfix P ij =15andΩ ij =5foralllinkssothatthecapacityoneach linkisdecidedonlybyitsnoiseintensity. Figure3.4displaystimeslotevolutionoftotalqueuelengthforHDwithβ =0andforthe original BP (V = 0). Average queue congestion is minimized at about 50 packets under HD, compared with over 100 packets under original BP. Further, little steady-state oscillations in total queue congestion under HD contrary to its large variations under BP verifies H4 in the previoussubsection. In minimizing average routing cost, Fig. 3.5 displays timeslot evolution of total power consumption for Pareto optimal HD withβ = 1 and for V-parameter BP withV = 10. Note thatwhilethetotalpowerconsumptionandtheaverageroutingcostarenotidentical,theyare highly correlated with each other. Smaller steady-state oscillations in total power under HD endorses both H1 and H4 in the previous subsection, showing the defect of link capacity-driven scheduling andmaximumpacketforwardingbyBP. 20 40 60 80 100 120 0 500 1000 1500 Total queue length BP (V=0) HD ( β=0) d 1 2 3 4 timeslot Fig. 3.4 Power minimization: Timeslot total queue backlog in HD withβ = 0 versus original BP, showing the minimization of average queue congestion by HD. Noticeable is also the little steady-state oscillations in total queueunderHDcontrarytoitslargevariationsunderBP. 3.3UniclassHeat-Diffusion(HD)Policy 45 0 100 200 300 0 20 40 60 80 timeslot Total power usage BP (V=10) HD ( β=1) Fig.3.5 Power minimization: Timeslottotalpowerconsumption, whichishighlycorrelated with theDirichletroutingcost,inParetooptimalHDwithβ = 1versusV-parameterBPwithV = 10. Figure3.6displaysthetradeoffbetweenqueuecongestionandpowerusageinParetooptimal HDasafunctionofβ andinV-parameterBPasafunctionofV. TheresultsverifyH3inthe previous subsection and concur with the graphical illustration of HD Pareto optimality depicted by Fig. 3.1. Theyalso match thetimeslot evolutionresults displayedin Fig.3.4 fortotal queue lengthatβ =0andV =0,andinFig.3.5fortotalpowerconsumptionatβ =1andV =10. Note the rapid growth of queue lengths in V-parameter BP when average power usage is pushed downwards, indicating the fact that the V-parameter BP cannot reach the minimum routing cost subjecttonetworkstability,i.e.,boundedqueuelengths. 200 400 600 800 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 200 400 600 800 0 100 200 300 0 0.2 0.4 0.6 0.8 1 Average total power usage Average total queue length β β 20 40 60 80 0 10 20 30 0 0.2 0.4 0.6 0.8 1 1 2 3 q HD BP V V 0 2 4 6 8 10 0 2 4 6 8 10 20 40 60 80 0 10 20 30 0 0.2 0.4 0.6 0.8 1 1 2 3 q HD BP Fig. 3.6 Power minimization: Queue congestion versus power consumption in Pareto optimal HD againstβ and in V-parameter BP againstV,withthedashedlinesrepresentinginterpolation. 46 Heat-Diffusion: ParetoOptimalDynamicRouting 3.4 KeyPropertyofPareto Optimality Considerageneraluniclassqueuingnetworkwithasingledestinationnoded. Asbefore,let q i (n) be the number of existing packets at nodei at slotn. State variables of the system can thenberepresentedbythefollowingvector: q ◦ (n):= q 1 (n),...,q d− 1 (n),q d+1 (n),...,q |V| (n) . Notethatq d (n)≡ 0 isdiscardedfromstatevariables. Notation 3.1. We use subscript◦ to denote reduced vectors or matrices obtained by discarding theentriescorrespondingtothedestinationnoded. Let a stochastic processa i (n) represent the number of exogenous packets arriving into node i atslotn. Discarda d (n)≡ 0 andcomposethevectorofnodearrivalsas a ◦ (n):= a 1 (n),...,a d− 1 (n),a d+1 (n),...,a |V| (n) . Alsocomposethevectoroflinkactualtransmissionsas f(n):= f 1 (n),...,f |E| (n) where,asbefore,f ij (n) representsthenumberofpacketsactuallysentoverlinkij atslotn. Givenadirectedgraph(V,E), letB denotethenode-edgeincidencematrix,inwhichB iℓ is 1 if nodei is the tail of directed edgeℓ,− 1 ifi is the head, and 0 otherwise. 1 ThenB ◦ denotes a reduction ofB that discards the row related to the destination noded, which is referred to asreducedincidencematrix. OnecanverifythatB ◦ f(n)isanodevector,inwhichtheentry correspondingtonodei reads (B ◦ f) i (n)= X b∈out(i) f ib (n)− X a∈in(i) f ai (n). Using the above notation, thef-controlled, stochastic state dynamics of a uniclass queuing networkiscapturedby q ◦ (n+1)=q ◦ (n)+a ◦ (n)− B ◦ f(n). (3.10) 1 Referring to Sec. 2.2.6, one can view graph as a 1-complex, whereB is its 1-incidence matrix that describes thecorrelationbetweenalloriented1-cells(edges) and 0-cells (nodes) in the complex. 3.4KeyPropertyofParetoOptimality 47 Notethatthelinkcapacitiesµ ij (n)varybychannelstates,whilethelinkactualtransmissions f ij (n) are assigned by a routing policy subject to 0 6 f ij (n) 6 min{q i (n), µ ij (n)}. This difference explains why despite traditional notation in literature, there is no need for (·) + operationinthequeueequation (3.10). In the wake of (3.10), the next theorem formalizes the HD main characteristic, which is centraltotheproofofTh.3.2onHDthroughput-optimality,Th.3.3onHDaveragenetwork delay minimization, Th. 3.5 on connection between HD fluid limit and combinatorial heat equation, and Th. 3.8 on HD average quadratic routing cost minimization. Before proceeding to thetheorem,letusdefinethelinkweightmatrixas Φ (n):=diag(φ (n)) (3.11) whereφ (n) representsthevectorcomposedofφ ij (n) asdefinedin (3.6). Theorem 3.1. (HD Key Property) At every timeslotn and for allβ ∈ [0,1], Pareto optimal HDpolicymaximizesthef-controlledfunctional D(f,q ◦ ,n):=2f(n) ⊤ Φ (n)B ◦ ⊤ q ◦ (n)− f(n) ⊤ f(n) (3.12) subjecttonetworkconstraints,includingdirectionality,capacityandinterference. Proof. Onecanverifythat D(f,q ◦ ,n)= X ij∈E 2φ ij (n)q ij (n)f ij (n)− f ij (n) 2 . Letustemporarilyrelaxallnetworkconstraints. Theneachlink-relatedcomponentofD(f,q ◦ ,n) turnstobestrictlyconcave. Foreachlinkij,bytakingthefirstderivativewithrespectto f ij , wefindthemaximizinglinktransmission f opt ij =φ ij q ij . Consideringthelinkconstraintsthatf ij mustbenon-negativeandatmostequaltothelinkcapacityresultsinf opt ij =min{φ ij q ij + , µ ij } which follows ̂ f ij in (3.6). Considering the link interference constraint, on the other hand, enforcestoactivatethelinksthatcontributemosttotheD maximization. Thenassumingthat aninterference model does not let anode transmitto morethanone neighboratthe same time, thelatterdirectlyleadstothemax-weightscheduling (3.8)alongsidetheHDweighting (3.7), whichconcludestheproof. 48 Heat-Diffusion: ParetoOptimalDynamicRouting Considerthelong-termaverageoffunctionalD(f,q ◦ ,n) definedas D(f,q ◦ ):=2f ⊤ Φ B ◦ ⊤ q ◦ − f ⊤ f. (3.13) Next assumption is being used in the analytical proofs of HD properties, stating that the greedy maximization ofD(f,q ◦ ,n) at each timeslot leads to its maximum long-term average. The assumptionimpliesthatonecanapplytheBellman’sprincipleofoptimality,andsodynamic programming, to maximize D. It also implicitly means no overlapping among slot-based substructuresofD maximizationproblem. Assumption3.1. Givenacombinationofnetworktopologyandtrafficrates,timeslotmaximiza- tion ofD(f,q ◦ ,n) is an optimal substructure for global maximization ofD(f,q ◦ ), subject to networkconstraints. In practice, almost every wireless mesh network meets this assumption. As an example that fails the requirement though, consider the case where exogenous packets arrive only to onenode,saya,whichisconnecteddirectlytothefinaldestination. Assumethatalllinksare bidirectionalwithunitcostfactorsandinfinitecapacities,andsolinkinterferenceistheonly network constraint. Obviously, depleting the whole queue into the destination maximizesD to q a (n) 2 at each timeslot. To maximize D, however, a portion of traffic must be forwarded throughotherpathsthatconnectnodea tothedestination. 3.5 HDThroughputOptimality Let the stochastic process S(n) = S 1 (n),··· ,S |E| (n) represent channel states at slot n, describing all uncontrollable factors that affect wireless link capacities and cost factors. We assume thatS(n) evolves according to an ergodic stationary process and takes values in a finite setS. Thus,byBirkhoff’sergodictheorem,eachstate S∈S hasaprobabilityof s:=P S(n)=S =limsup τ →∞ 1/τ X τ − 1 n=0 I S(n)=S (3.14) where P S∈S s=1. Thentheexpectedlinkcapacitiesandcostfactorsareobtainedas E µ (n)}= X S∈S sE µ (n) S(n)=S (3.15) E ρ (n)}= X S∈S sE ρ (n) S(n)=S (3.16) 3.5HDThroughputOptimality 49 whereµ (n) andρ (n) represent the vectors composed of link capacitiesµ ij (n) and link cost factorsρ ij (n),respectively,atslotn. Notethattheexistenceofprobabilitydistribution(3.14)orexpectedvalues(3.15)and(3.16) bynomeansimplythattheyareknowntoaroutingpolicy. Specifically,HDperformswithno needofknowinganyoftheseinformation. Nonetheless,theergodicityofS(n)alongwiththe lawoflargenumbersimply E µ (n)}= lim τ →∞ 1/τ X τ − 1 n=0 µ (n) E ρ (n)}= lim τ →∞ 1/τ X τ − 1 n=0 ρ (n) meaning that the expectations converge to the long-term averages. Thus, a routing policy could estimateE{µ (n)} andE{ρ (n)} by observing timeslot variables µ (n) and ρ (n) for a long enoughperiodoftime,atleastintheory. Thisjustifiestheexistenceofstationaryrandomized policiesthatbasetheirroutingdecisionsonlyonarrivalstatisticsandchannelstateprobabilities, butfullyindependentofqueueoccupancies. 3.5.1 Characteristicofnetworkcapacityregion Consider a uniclass wireless network that is described by a connectivity graph(V,E), a destina- tionnoded andanergodicstationarychannelstateprocessS(n). Definition 3.6. Given a routing policy, its stability region is the set of all arrival vectors that it canstablysupport,i.e.,makethenetworkstableunderthosearrivals. Definition 3.7. Given a network layer, its capacity region is the union of stability regions achievedbyallroutingpolicies,includingthosewhicharepossiblyunfeasible. It can be shown that for any network, its capacity region is always convex and compact and soisclosedandbounded[72]. Definition 3.8. A routing policy is throughput-optimal if it can stabilize the entire network capacityregion,i.e.,securequeuestabilityunderallstabilizablearrivalvectors. An arrival vectora ◦ (n) is in the network capacity region, i.e., stabilizable, if and only if thereexistsasetoflinkactualtransmissionsf(n) thatsatisfy a i = X b∈out(i) f ib − X a∈in(i) f ai , ∀i∈V\{d} (3.17) 50 Heat-Diffusion: ParetoOptimalDynamicRouting constrained by link capacities and interference. Under an ergodic channel state process, this basicallyreadsthelong-termaverageflowconservationatthenodes. Inamatrixform, (3.17) canequivalentlybeshownbya ◦ =B ◦ f. Remark 3.3. Linkactualtransmissionsf(n)arenotfixed,butdependonroutingpolicy. Fur- ther, there could potentially exist infinite number of routing policies that meet (3.17) for any stabilizablea ◦ (n). Amongthemaretheonesthatusethesimpleprobabilityconceptofdistribut- ing packets randomly so that the desired time averages(3.17) can be achieved. These stationary randomizedpoliciesareprohibitiveinpracticeastheytypicallyrequireperfectknowledgeof arrival statistics and channel state probabilities along with an expensive computation. Nonethe- less, the fact that these queue-independent policies exist plays a crucial role in the analytical proofofHDpropertiesinthisandnextsection. 3.5.2 HDthroughputoptimalityforallβ ToprovenetworkstabilityunderParetooptimalHDpolicy,aswellassomeotherHDproperties in next sections, we are compelled to choose unorthodox Lyapunov candidates based on the followingnonsymmetricsystemmatrix: M ◦ (n):= B ◦ B ◦ ⊤ − 1 B ◦ Φ (n)B ◦ ⊤ . (3.18) HandlingLyapunovargumentsturnstobealotmorechallenging,sincetheeasywayofworking withsymmetricpositivedefinitematricesceasestoexisthere. Nonetheless,thespecificstructure ofM ◦ (n) makesthefollowinglemmaspossible. Lemma3.1. Givenaconnecteduniclasswirelessnetwork,M ◦ (n)ispseudopositivedefinite in the sense that all of its eigenvalues are positive andx ⊤ M ◦ (n)x>0 for any vectorx∈R |V|− 1 , withequalityifandonlyifx=0. Proof. We drop timeslot variable (n) for ease of notation. Let us define ∆ := B ◦ B ◦ ⊤ and ∆ φ := B ◦ Φ B ◦ ⊤ , which are both positive definite matrices as discussed in Sec. 2.3. Since ∆ 1/2 φ ∆ − 1 ∆ 1/2 φ iscongruentto∆ − 1 whichhasonlypositiveeigenvalues,bySylvester’slawof inertia,∆ 1/2 φ ∆ − 1 ∆ 1/2 φ hasonlypositiveeigenvaluestoo. ThelatterissimilartoM ◦ ,andsothey havethesameeigenvalues,provingthatM ◦ hasonlypositiveeigenvalues. 3.5HDThroughputOptimality 51 We now show thatx ⊤ M ◦ x> 0. Lettingv := ∆ − 1 x and substituting forM ◦ from the definition (3.18),itsufficestoshowthat (B ◦ ⊤ v) ⊤ Φ B ◦ ⊤ B ◦ (B ◦ ⊤ v)>0. (3.19) Doing another change of variable, let f := B ◦ ⊤ v that represents an edge vector in which f ij =v i − v j ,∀ij∈E. Recall thatB ◦ is a signed node-edge incidence matrix with arbitrarily assigned algebraic topological edge orientations. Let us assign edge orientations such that f ij > 0,∀ij ∈E. Then to fulfill (3.19), it suffices to show that f ⊤ Φ B ◦ ⊤ B ◦ f > 0 subject to f<0, which readsf ij >0,∀ij∈E. To this end, we equivalently show that minimum cost in thefollowingoptimizationproblemisnon-negative: min f<0 f ⊤ Φ B ◦ ⊤ B ◦ f. LetusconstructtheLagrangiandualproblem max λ <0 min f L(λ ,f):=f ⊤ Φ B ◦ ⊤ B ◦ f− λ ⊤ f (3.20) whereλ isthevectorofLagrangemultipliers. Becausetheprimalvariablef iscontinuously differentiable,sois L,andsotheminimumoccurswhere∇ f L=0,whichleadsto λ = B ◦ ⊤ B ◦ Φ +Φ B ◦ ⊤ B ◦ f opt . Substitutingf opt in (3.20)andnotingf opt andλ arebothentrywisenon-negative,weobtain max λ <0 L(λ ,f opt )=max λ <0 − 1 2 λ ⊤ f opt =0. (3.21) Bytheweakdualitytheorem,theminimumoftheprimalproblemisgreaterthanorequaltothe maximumofthedualproblem. Thus, (3.21)entailsmin f<0 f ⊤ Φ B ◦ ⊤ B ◦ f>0,whichequally meansx ⊤ M ◦ x>0. It remains to show thatx ⊤ M ◦ x=0 only ifx=0, which is equivalent to show that matrix M ◦ ⊤ +M ◦ ispositivedefinite. Since x ⊤ (M ◦ ⊤ +M ◦ )x>0alreadyguaranteesthatM ◦ ⊤ +M ◦ is positive semi-definite, it suffices to show that M ◦ ⊤ +M ◦ has no zero eigenvalue. Let us assume itdoes,whichimpliestheexistenceofaneigenvectorν ̸=0 suchthat (M ◦ ⊤ +M ◦ )ν =0. (3.22) 52 Heat-Diffusion: ParetoOptimalDynamicRouting BecauseM ◦ istheproductoftwopositivedefinitematrices, ν ̸=0entailsM ◦ ν ̸=0,which leadsto(M ◦ ν ) ⊤ M ◦ ν +(M ◦ ⊤ ν ) ⊤ M ◦ ⊤ ν >0. Utilizing(3.22)inthelatterresultsin ν ⊤ M ◦ ⊤ − M ◦ 2 ν <0 which is not true as (M ◦ ⊤ − M ◦ ) 2 is a symmetric positive semi-definite matrix. Therefore, M ◦ ⊤ +M ◦ hasnozeroeigenvalueandsoissymmetricpositivedefinite. Lemma 3.2. Given a connected uniclass wireless network, for any vectorx ∈ R |V|− 1 , the followingidentityholds: B ◦ ⊤ M ◦ (n)x=Φ (n)B ◦ ⊤ x. (3.23) Proof. We drop timeslot variable(n) for ease of notation. We haveB ◦ B ◦ ⊤ M ◦ x=B ◦ Φ B ◦ ⊤ x, which could easily be seen by substitutingM ◦ from (3.18). Thus, to prove the claim, it suffices toshowthatforanyvectorsxandy,equalityB ◦ y =B ◦ Φ B ◦ ⊤ xentailsy =Φ B ◦ ⊤ x. Tothis end,weutilizethepropertiesofheat equationsonundirectedgraphs(seeSec.3.7.2). Considerathermalgraphwithreducednode-edgeincidencematrixB ◦ andedgethermal diffusivity matrix Φ and let the destination node be fixed at zero temperature. As the first scenario, let us envisiony as the vector of heat fluxes through the branches, implying that B ◦ y represents the vector of heat sources injected at the nodes (see (3.44) and (3.45) under constant heat sources.) As the second scenario, envisionx as the vector of temperatures at the nodes, implyingthatΦ B ◦ ⊤ xrepresentsthevectorofheatfluxesthroughthebranchesand B ◦ Φ B ◦ ⊤ x representsthevectorofheatsourcesinjectedatthenodes. AssumingB ◦ y =B ◦ Φ B ◦ ⊤ x means that the thermal graph is charged by the same configu- rationofheatsourcesinbothscenariosabove. Itfollowsthatthevectoroftemperaturesatthe nodes are also the same as the Dirichlet Laplacian is positive definite in (3.45). Hence, in both scenarios the vector of heat fluxes through the branches must be equal, because B ◦ has full row rankin (3.44). Thisentailsy =Φ B ◦ ⊤ x,whichconcludestheproof. Lemma 3.3. Given a connected uniclass wireless network, there exists such a scalar16η 63 thatforanyvectorsx,y∈R |V|− 1 ,thefollowinginequalityholds: x ⊤ M ◦ (n) ⊤ +M ◦ (n) y6η x ⊤ M ◦ (n)y. (3.24) Proof. We drop timeslot variable (n) for ease of notation. Let us replace M ◦ +M ◦ ⊤ by 2M ◦ +(M ◦ ⊤ − M ◦ ). Doing some matrix manipulation, we need to show that there exists such 3.5HDThroughputOptimality 53 16η 63 thatforarbitraryvectorsx andy, x ⊤ (M ◦ ⊤ − M ◦ )y6(η − 2)x ⊤ M ◦ y. (3.25) Tothisend,itsufficestoshow x ⊤ (M ◦ ⊤ − M ◦ )y 6 x ⊤ M ◦ y ,whichthenmakestheinequality (3.25) true forη = 1 in case ofx ⊤ M ◦ y6 0, and forη = 3 in case ofx ⊤ M ◦ y > 0. This is equivalenttoshowthatthefollowinginequalityholds: x ⊤ (M ◦ ⊤ − M ◦ )yy ⊤ (M ◦ − M ◦ ⊤ )x6x ⊤ M ◦ yy ⊤ M ◦ ⊤ x. Bylittlealgebra,thelattercanberephrasedas x ⊤ (2M ◦ − M ◦ ⊤ )yy ⊤ M ◦ x>0. To prove the above inequality, it suffices to show that the minimum objective value in the followingoptimizationproblemisnon-negative: min x,y x ⊤ (2M ◦ − M ◦ ⊤ )yy ⊤ M ◦ x s.t. x ⊤ M ◦ x>0 , y ⊤ M ◦ y >0 where the constraints are enforced in light of Lem. 3.1. The Lagrangian dual problem, withλ x andλ y astheLagrangemultipliers,isfoundas max λ x,λ y>0 min x,y L:=x ⊤ (2M ◦ − M ◦ ⊤ )yy ⊤ M ◦ x− λ x x ⊤ M ◦ x− λ y y ⊤ M ◦ y Imposingthefirstorderconditions ∇ x L=0 and∇ y L=0 leadsto λ x (M ◦ ⊤ +M ◦ )x=2M ◦ yy ⊤ M ◦ x+2M ◦ ⊤ yy ⊤ M ◦ ⊤ x− 2M ◦ ⊤ yy ⊤ M ◦ x λ y (M ◦ ⊤ +M ◦ )y =2M ◦ xx ⊤ M ◦ y+2M ◦ ⊤ xx ⊤ M ◦ ⊤ y− 2M ◦ xx ⊤ M ◦ ⊤ y. Let us plug these two equations into the LagrangianL and utilize the identitiesx ⊤ M ◦ ⊤ y = y ⊤ M ◦ x:=a andx ⊤ M ◦ y =y ⊤ M ◦ ⊤ x:=b witha andb being scalars. One can easily confirm thatL=(2ab− b 2 )− λ x x ⊤ M ◦ x− λ y y ⊤ M ◦ y andλ x x ⊤ (M ◦ ⊤ +M ◦ )x=2(2ab− b 2 )and λ y y ⊤ (M ◦ ⊤ +M ◦ )y =2(2ab− b 2 ). Thenbylittlealgebra,theLagrangiantransformsto L= 1 4 λ x x ⊤ M ◦ ⊤ − 3M ◦ x+ 1 4 λ y y ⊤ M ◦ ⊤ − 3M ◦ y. 54 Heat-Diffusion: ParetoOptimalDynamicRouting ObservethatM ◦ − M ◦ ⊤ isaskew-symmetricmatrix,andsobothx ⊤ (M ◦ − M ◦ ⊤ )xandy ⊤ (M ◦ − M ◦ ⊤ )y vanish. ThentheLagrangiandualproblemreads max λ x,λ y>0 L= max λ x,λ y>0 − 1 2 λ x x ⊤ M ◦ x+λ y y ⊤ M ◦ y =0. This entailsx ⊤ (2M ◦ − M ◦ ⊤ )yy ⊤ M ◦ x>0 by the weak duality theorem that the maximum of thedualproblemprovidesalowerboundfortheminimumoftheprimalproblem. ToanalyzetheHDthroughputoptimality,considertheLyapunovcandidate W(n):=q ◦ (n) ⊤ M ◦ (n)q ◦ (n). Though W(n) is indeed an energy function in light of Lem. 3.1, due to the nonsymmetric weighting matrixM ◦ (n), it has no trivial interpretation of a specific energy in the system. Nonetheless,itclearlypenalizeshighqueuedifferentialsacrosslinks,compellingamoreeven distribution of packets over the network. It also incites transmission over the links of lower cost factors, leading to a less expensive routing decision. Note that either atβ =0 or for the case that all links are of the same cost factor,Φ (n) is simplified to a scaled identity matrix that leads toM ◦ (n) = Φ (n), which in turn reduces W(n) to the sum of squares of queue lengths—a familiarLyapunovfunctioninmostofpreviousresultsinliterature. Let ∆ W(n) := W(n + 1)− W(n) be the Lyapunov drift. Substituting for q ◦ (n + 1) from (3.10)leadsto ∆ W(n)= a ◦ (n)− B ◦ f(n) ⊤ M ◦ (n)+M ◦ (n) ⊤ q ◦ (n) + a ◦ (n)− B ◦ f(n) ⊤ M ◦ (n) a ◦ (n)− B ◦ f(n) . Let us drop timeslot variable(n) for ease of notation. Applying Lem. 3.3 to the first line in the aboveLyapunovdriftequationyields ∆ W 6η (a ◦ − B ◦ f) ⊤ M ◦ q ◦ +(a ◦ − B ◦ f) ⊤ M ◦ (a ◦ − B ◦ f) with16η 63. Letusreplacef ⊤ B ◦ ⊤ M ◦ q ◦ byf ⊤ Φ B ◦ ⊤ q ◦ inlightofLem.3.2,addandsubtract theterm 1 2 η f ⊤ f,andusetheD(f,q ◦ ,n) expressionin (3.12)toobtain ∆ W 6η a ◦ ⊤ M ◦ q ◦ − η 2 D(f,q ◦ ,n)− η 2 f ⊤ f +(a ◦ − B ◦ f) ⊤ M ◦ (a ◦ − B ◦ f). 3.5HDThroughputOptimality 55 Takingconditionalexpectationgiventhecurrentqueuebacklogsq ◦ (n)andknowingthatthe termη f ⊤ f hasazerolowerbound leadto E ∆ W|q ◦ 6η E a ◦ ⊤ M ◦ q ◦ q ◦ − η 2 E D(f,q ◦ ) q ◦ + E (a ◦ − B ◦ f) ⊤ M ◦ (a ◦ − B ◦ f) q ◦ (3.26) wheretheconditionalexpectationiswithrespecttotherandomnessofarrivals,channelstates androutingdecision—incaseofarandomizedroutingalgorithm. Observe thatM ◦ (n) = B ◦ B ◦ ⊤ − 1 B ◦ Φ (n)B ◦ ⊤ is a function only of control parameterβ andlinkcostfactorsρ ij (n). Sincearrivalsareindependentofbothβ andρ ij ,weget E a ◦ ⊤ M ◦ q ◦ =E a ◦ ⊤ q ◦ E M ◦ q ◦ . Atthesame time, bothβ andρ ij areindependentofq ◦ ,soisM ◦ ,whichmeansE{M ◦ |q ◦ }= E{M ◦ }. Ontheotherhand,sincethenetworklayerroutingcontrollerhasnoimpactonarrivals, a ◦ (n) turns to be an independent system variable that is not influenced by anything, which impliesE{a ◦ ⊤ |q ◦ }=E{a ◦ ⊤ }. Puttingtheseresultstogetheryields E a ◦ ⊤ M ◦ q ◦ q ◦ =E{a ◦ ⊤ }E{M ◦ }q ◦ . (3.27) Giventhecurrentqueuebacklogsq ◦ (n),letf ⋆ (n)bethelinkactualtransmissionsprovided by HD policy. As compared to any alternative transmission decisionf(n), Th. 3.1 secures D(f ⋆ ,q ◦ ,n)>D(f,q ◦ ,n)forallβ andateachslotn. Consideringthiswiththeequality(3.23) ofLem.3.2implies D(f ⋆ ,q ◦ ,n)>2f ⊤ B ◦ ⊤ M ◦ q ◦ − f ⊤ f. Takingconditionalexpectationgivencurrentqueuebacklogsleadsto E D(f ⋆ ,q ◦ ,n) q ◦ >2E f ⊤ B ◦ ⊤ M ◦ q ◦ q ◦ − E f ⊤ f q ◦ . Asonealternativetransmissiondecisionf(n)tobecomparedwiththef ⋆ (n)providedby HD policy, consider the case wheref(n) is produced by a routing algorithm which makes inde- pendent, stationary and randomized transmission decisions at each slotn based only on arrivals andlinkcapacities. Theexistenceofsuchanalgorithm,whichmakesdecisionsindependent ofbothqueuebacklogsandlinkcostfactors,isknown[72]. Letusfix f(n)forthisalgorithm and refer to it asf ′ (n). Using equalityE{M ◦ |q ◦ } =E{M ◦ } and considering thatf ′ (n) is 56 Heat-Diffusion: ParetoOptimalDynamicRouting independentfromq ◦ (n) andM ◦ (n), weobtain E D(f ⋆ ,q ◦ ,n) q ◦ >2E{f ′⊤ B ◦ ⊤ }E{M ◦ }q ◦ − E{f ′⊤ f ′ }. Exploitingthisand (3.27)in (3.26)leadstothefollowingLyapunovdriftinequalitywhichis evaluatedunderHDpolicygivencurrentqueuebacklogsatslotn: E ∆ W|q ◦ 6η E{(a ◦ − B ◦ f ′ ) ⊤ }E{M ◦ }q ◦ +E Γ |q ◦ Γ:=( a ◦ − B ◦ f ⋆ ) ⊤ M ◦ (a ◦ − B ◦ f ⋆ )+ η 2 f ′ ⊤ f ′ . InvestigatingΓ( n), note that (i) all arrivals are of finite mean and variance, (ii) each link actual transmissionis atmostequaltothelinkcapacitywhichisfinite,andsoboth f ⋆ (n) and f ′ (n)havefiniteupperbounds,and(iii) M ◦ (n)isapseudopositivedefinitematrixinthesense of Lem. 3.1 with finite entries (recall φ ij (n)61). Thus, the expected value ofΓ( n) is finite at each slotn, and so there exists a finite positive scalar Γ max such thatE{Γ( n)|q ◦ (n)}6 Γ max . UtilizingthisintheLyapunovdriftinequalityyields E ∆ W|q ◦ 6η E{(a ◦ − B ◦ f ′ ) ⊤ }E{M ◦ }q ◦ +Γ max . (3.28) Inthewakeof (3.28),thenexttheoremisprovenbyshowingthatE{∆ W|q ◦ }isalwaysnegative forallβ ∈[0,1]. Theorem 3.2. (HDThroughputOptimality)Onuniclasswirelessnetworks,Paretooptimal HDpolicywithanyβ ∈[0,1]isthroughput-optimal,meaningthatitguaranteesnetworkstability underallstabilizablearrivalvectors. Proof. To simplify the rest of the proof, we assume arrivals to be i.i.d. over timeslots. For non-i.i.d. arrivals with stationary ergodic processes of finite mean and variance, a similar analysis can be done using N-slot Lyapunov drift [72], where the queue evolution (3.10) is modifiedto q ◦ (n+N)=q ◦ (n)+ X n+N− 1 k=n a ◦ (k)− X n+N− 1 k=n B ◦ f(k). (3.29) One can viewN as the time required for the system to reach “near steady state,” noting that in thei.i.d. case,thesteadystateisreachedoneachandeverytimeslot,andsoN =1. We often drop timeslot variable(n) for ease of notation. Back to the proof for i.i.d. arrivals, supposethata ◦ isinteriortothenetworkcapacityregionC. Thus,thereexistsanϵ > 0such 3.5HDThroughputOptimality 57 thata ◦ +ϵ 1∈C. Sincethestationaryrandomizedalgorithmthatgeneratesf ′ (n)isthroughput optimal [72], it can stabilize the arrivala ◦ +ϵ 1 at each timeslot. The i.i.d. assumption on arrivalsandchannelstatesthenleadsto E{a ◦ − B ◦ f ′ }=a ◦ − (a ◦ +ϵ 1)=− ϵ 1 implying that botha ◦ andf ′ reach their steady states on each and every timeslot. Plugging the latterintotheLyapunovdriftinequality (3.28)yields E ∆ W|q ◦ 6− ηϵ 1 ⊤ E{M ◦ }q ◦ +Γ max . (3.30) Letusassumethatthereexistsaµ> 0,whichisexploredlater,suchthat1 ⊤ E{M ◦ }q ◦ >µ 1 ⊤ q ◦ . Usingthisinthelatterdriftinequalityleadsto E ∆ W|q ◦ 6− ηµϵ 1 ⊤ q ◦ +Γ max . Thus,E{∆ W|q ◦ }<0forany P i q i >Γ max (ηµϵ ). TheninlightofTheorem2in[51],the queuingsystemisstable,andsoa ◦ isintheHDstabilityregion. Thisimpliesthatanyarrival ratea ◦ being interior to the network capacity region is stabilized by HD with anyβ ∈ [0,1], meaningthatHDisthroughput-optimalforallβ ∈[0,1]. We now show that there exists such a µ> 0 that satisfies 1 ⊤ E{M ◦ }q ◦ > µ 1 ⊤ q ◦ . Let us temporarily ignore the expectation and solve the problem forM ◦ . The claim is trivial for q ◦ = 0, and so we assumeq ◦ ̸= 0. Further,q ◦ represents the vector of queue occupancies onthewirelessnetworkthatarealwaysnon-negative. Let∥q ◦ ∥ 1 representtheℓ 1 normofq ◦ , defined as the sum of all queue occupancies. With no loss of generality, one may normalize ∥q ◦ ∥ 1 toone. Theproblemcanthenberephrasedasfindinga µ> 0 suchthat min ∥q ◦ ∥ 1 =1,q ◦ <0 1 ⊤ (M ◦ − µ I)q ◦ >0. The latter is a standard linear programming problem. Using simplex method, the minimum lies on a vertex of the simplex, where the vertices of the simplex are the natural basis elements e j :j =1,...,|V|. Thus,theµ istobesoughtsuchthat 1 ⊤ (M ◦ ) :,j − µ e j >0. Thisimmediatelyleadstoµ =min j 1 ⊤ M ◦ e j . 58 Heat-Diffusion: ParetoOptimalDynamicRouting Itremainstoshowthat1 ⊤ M ◦ e j >0foreverynaturalbasise j . ByLem.3.3,thereexists sucha16η 63 as η 1 ⊤ M ◦ e j >1 ⊤ M ◦ ⊤ +M ◦ e j whichimplies(η − 1)1 ⊤ M ◦ e j >1 ⊤ M ◦ ⊤ e j . Therighthandsideisalwayspositivebythenext electricalcircuitargument,whichimplies(η − 1)1 ⊤ M ◦ e j >0. Thelatterentailsη > 1and 1 ⊤ M ◦ e j >0 aswedesired. Toarguethat1 ⊤ M ◦ ⊤ e j =e ⊤ j M ◦ 1>0,bysubstitutingM ◦ from (3.18),weneedtoshow e ⊤ j (B ◦ B ◦ ⊤ ) − 1 (B ◦ Φ B ◦ ⊤ )1>0. Letusassociatenode-edgeincidencematrixB ◦ witharesistivenetworkande j withthevectorof independentcurrentsourcesattachedtothenodes. Thenthevectorv :=(B ◦ B ◦ ⊤ ) − 1 e j readsthe voltagesinducedatthenodes. Sincee j impliesthatelectricalcurrentisinjectedintothenetwork byasinglecurrentsourceatthenodej,theresultingvoltageateachnodeisnon-negative(v<0). Further,thevoltagesatthenodej andatleastatoneofthenodesneighbortoground(destination node)arealwayspositive. Ontheotherhand,theelementsofeachrowoftheDirichletLaplacian B ◦ Φ B ◦ ⊤ sumtozero,exceptforthoserowsrepresentingthenodesneighbortoground,which alwayssumtoapositivevalue. (RecallthatB ◦ isobtainedfromB bydiscardingtherowrelated to ground.) This implies that in the vectoru := (B ◦ Φ B ◦ ⊤ )1, the components related to the nodesneighbortogroundarepositive,andothersarezero. Consideringtheconditionsofuand v together, wegete ⊤ j M ◦ 1=v ⊤ u>0. ReplacingM ◦ byE{M ◦ }, the same argument leads toµ =min j 1 ⊤ E{M ◦ }e j >0, which concludestheproof. 3.6 HDMinimumDelay atβ = 0 ParetooptimalperformanceofHDpolicystandsontwopillars: minimizationoftheaverage queue congestionQ withβ = 0, and minimization of the average routing costR withβ = 1. This section settles the first pillar based on a timeslot analysis. Let us start with two lemmas thathelpusprovethenetworkdelay minimizationresultinTh.3.3. Lemma 3.4. At β = 0 and subject to network constraints, timeslot maximization of the functionalD(f,q ◦ ,n) in (3.12)isequivalenttotimeslotmaximizationof G(f,q ◦ ,n):=2f(n) ⊤ B ◦ ⊤ q ◦ (n)− f(n) ⊤ B ◦ ⊤ B ◦ f(n). (3.31) 3.6HDMinimumDelayatβ =0 59 Proof. Letustemporarilyrelaxallnetworkconstraints. ThenG(f,q ◦ ,n)turnstobestrictly concave and so reaches its maximum at thef that meets the first order condition ∇ f G = 0. This leads toB ◦ ⊤ B ◦ f(n) =B ◦ ⊤ q ◦ (n), which is equivalent toB ◦ f(n) =q ◦ (n) asB ◦ is full rowrank(seeSec.2.3). Letg ij (n)denotethecomponentofvectorB ◦ ⊤ B ◦ f(n)correspondingto edgeij,i.e.,g ij (n):=(B ◦ ⊤ B ◦ f) ij (n). Thenthemaximizingf(n)mustmeetg ij (n)=q ij (n) oneachlinkij. Assume,ontheotherhand,thataroutingpolicytakesf ij (n)=α ij q ij (n)for ascalarα ij >0,whichwillbedeterminedlater. Followingourpreviousargument,forsucha policytomaximizeG,itmustcomply withf ij (n)=α ij g ij (n)ateachslotn. Expressingina matrix form, the latter readsf(n) = diag(α )B ◦ ⊤ B ◦ f(n)withα being the vector formed by α ij ’s. PluggingthisintoG(f,q ◦ ,n),themaximumisfoundbysolving max α ,f(n) 2f(n) ⊤ B ◦ ⊤ q ◦ (n)− f(n) ⊤ diag(α ) − 1 f(n). (3.32) Thendeployingf ij (n)=α ij q ij (n) in (3.32)leadsto G max (n)=max α q ◦ (n) ⊤ diag(α )q ◦ (n) whichrepresentsthemaximumofG withnonetworkconstraint. Inlightof(q ◦ − B ◦ f) ⊤ (q ◦ − B ◦ f)>0,weobtainG max (n)=q ◦ (n) ⊤ q ◦ (n),whichmeans the maximizingα ij is equal to one for all edges. Fixingα ij =1 and adding network constraints to the problem, it is obvious that theG max (n)=q ◦ (n) ⊤ q ◦ (n) is no longer attainable. However, going one step back to (3.32), we can find the maximizing f(n) by solving the following optimizationproblemateachslotn: max f(n) 2f(n) ⊤ B ◦ ⊤ q ◦ (n)− f(n) ⊤ f(n) s.t. Network Constraints. Observe that the objective function here is identical toD(f,q ◦ ,n) in Th. 3.1 for the case of φ ij (n)=1,orequivalentlyforβ =0,whichismaximizedbyHDateachtimeslot,subjectto network constraints. Thus, by the same way,G(f,q ◦ ,n) is maximized by HD withβ = 0 at eachtimeslot,subjecttonetworkconstraints. It is critical to understand that Lem. 3.4 does not claim about the same maximum values forfunctionalsD andG,whichisobviouslynottrue,butaboutthesamemaximizingcontrol actionf(n)ateachslotn. Anotherpointisthatwhileateachtimeslot,HDmaximizesD forall β ∈[0,1],itmaximizesG foronlyβ =0. 60 Heat-Diffusion: ParetoOptimalDynamicRouting Lemma3.5. Considerauniclasswirelessnetworkunderanarrivalratea ◦ thatisstabilizedbya routing policy, resulting in average queue occupanciesq ◦ and average link actual transmissions f . Thenthefollowingidentityholds: 2Cov{B ◦ f,q ◦ }− Var{B ◦ f}=2Cov{a ◦ ,q ◦ − B ◦ f}+Var{a ◦ } (3.33) whereVar{X}:=Cov{X,X}fortworandomvariablesX andY ,with Cov{X,Y}:=E{X ⊤ Y}− E{X} ⊤ E{Y}. Proof. We occasionally drop timeslot variable (n) for ease of notation. Consider W(n) := q ◦ (n) ⊤ q ◦ (n)astheclassicalquadraticLyapunovcandidateandtakeexpectationfromtheLya- punovdrift∆ W(n)=W(n+1)− W(n) toobtain E{∆ W}=E{a ◦ − B ◦ f} ⊤ E{a ◦ − B ◦ f+2q ◦ } − 2Cov{B ◦ f,q ◦ }+Var{B ◦ f} +2Cov{a ◦ ,q ◦ − B ◦ f}+Var{a ◦ } (3.34) wherethe equality holdsateachtimeslot andexpectationiswithrespect totherandomnessof arrivals,channelstatesand(possibly)routingdecision. Letg :=a ◦ − B ◦ f +2q ◦ ,sumover timeslots0untilτ − 1,dividebyτ andtakealimsupofτ →∞frombothsidesof (3.34)to obtainthefollowingexpectedtimeaverageequation: limsup τ →∞ 1 τ X τ − 1 n=0 E{a ◦ (n)− B ◦ f(n)} ⊤ E{g(n)}= +2Cov{B ◦ f,q ◦ }− Var{B ◦ f} − 2Cov{a ◦ ,q ◦ − B ◦ f}− Var{a ◦ } (3.35) where we utilizedlimsup τ →∞ (W(τ )− W(0))/τ = 0 as the routing policy stabilizesa ◦ and sokeepsW(n) finitewithprobability1ateachtimeslot. Itremainstoshowthattheleft-handsideof (3.35)vanishes. Observethatg(n)isentrywise non-negativeandfinite. Thus,thereexistconstantvectors g min andg max suchthat 04g min 4E{g(n)}4g max . 3.6HDMinimumDelayatβ =0 61 Hence,theleft-handsideof (3.35)isboundedfrombelowto(a ◦ − B ◦ f) ⊤ g min andfromabove to(a ◦ − B ◦ f) ⊤ g max . Further,asa ◦ isstabilizedbytheroutingpolicy,thefeasibilitycondition in (3.17)entailsa ◦ =B ◦ f,implyingthattheleft-handsideof (3.35)vanishes. To gain an insight into this lemma, consider a constant arrival vector which makes the right- hand side of (3.33) vanished. In light ofCov{B ◦ f,q ◦ } = Cov{f,B ◦ ⊤ q ◦ }, equality (3.33) then implies that a stabilizing routing decision with a higher average total variance of link forwardingsnecessarilyresultsinahigheraveragetotalcovariancebetweenlinkforwardings andlinkqueue-differentials. Forexample,comparedwithBPthatsaturatesactivatedlinksto theircapacitylimits,HDwithamoreconservativepacketforwardingresultsinlessvariations in link actual transmissions. The lemma then claims that HD leads to a smaller correlation betweenlinkforwardingsandlinkqueue-differentials,whichisconfirmedbycomparingHD andBPalgorithms(seeH4inSec.3.3.2). Definition 3.9. We specifyD-class routing policies as a collection of all dynamic routing policies that make timeslot routing decisions based only on current queue occupancies and channel statesandsoindependent ofarrivalstatisticsandchannelstateprobabilities. By allowing as many routes as possible,D-class routing policies tend to distribute traffic all over the network. This class includes all opportunistic max-weight schedulers that do not incor- porate the Markov structure of topology process into their routing decisions, including original BP [44] and most of its derivations [45–50, 52–59, 61, 62, 64]. The class also encompasses all offlinestationaryrandomizedalgorithms(possiblyunfeasible)thatmakeroutingdecisionsas purefunctionsonlyofobservedchannelstates,andsoindependentofqueueoccupancies,by typicallyusingtheknowledgeofarrivalstatisticsandchannelstateprobabilities. Theorem 3.3. (HD Minimum Delay) Consider a uniclass wireless network that meets As- sum. 3.1 under a stabilizable arrival rate. WithinD-class routing policies, Pareto optimal HDwithβ = 0 minimizestheaverage totalqueuecongestionQ as definedin (3.2), whichis proportionaltoaveragenetworkdelaybyLittle’sTheorem. Proof. Tosimplifytheproof,weassumearrivalsarei.i.d. overtimeslots,withtheunderstanding that it can easily be modified to yield similar result for non-i.i.d. arrivals, using the N-slot analysisderivedfrom (3.29). Weoftendroptimeslotvariable(n) foreaseofnotation. Consider an arrival ratea ◦ interior to the stability region of aD-class routing policy, which werefertoitas“generic”. Letthetimeslotquantitiesf(n)andq ◦ (n)beproducedbysucha generic routing policy. If this generic routing policy also maximizes theG functional (3.31) at 62 Heat-Diffusion: ParetoOptimalDynamicRouting each slotn, by Assum. 3.1, it will result in the sameQ as that of HD policy atβ =0. Thus, we assumetheGobtainedbythegenericpolicyisnotmaximal. Thenforasufficientlysmall ϵ> 0, thereexistsa routingalgorithm(possiblyunfeasible)thatcanstabilizethearrivala ◦ +ϵ 1 while makingG(f,q ◦ ,n)notlessthanthatofthegenericroutingpolicyateachslotn. Letusreferto thisalgorithmas“fictitious,”aswedonotintendtoknowhowitreallyworks. Torestassure thatsuchanalgorithmexists,onemayendowitwiththeabilityofperfectlypredictingallfuture eventswithnouncertainty. Letf ′ (n) represent the vector of link actual transmissions produced by the fictitious al- gorithm at slotn givenq ◦ (n). Let us take expectation fromG(f ′ ,q ◦ ,n)> G(f,q ◦ ,n) with respect to the randomness of arrivals, channel states and routing decision. Due to the flow feasi- bilityconditionin(3.17)andthei.i.d. assumptiononarrivals,wegetE{B ◦ f ′ }=E{B ◦ f}+ϵ 1. UtilizingthelatterinE{G(f ′ ,q ◦ ,n)>G(f,q ◦ ,n)},weobtain 2ϵ 1 ⊤ E{q ◦ }> 2ϵ 1 ⊤ E{B ◦ f ′ }− ϵ 2 1 ⊤ 1+ 2Cov{B ◦ f,q ◦ }− Var{B ◦ f} − 2Cov{B ◦ f ′ ,q ◦ }− Var{B ◦ f ′ } whichholdsforeachtimeslot. Summingovertimeslots0untilτ − 1,dividingbyτ andtakinga limsupofτ →∞frombothsidesleadtothefollowingexpectedtimeaverageinequality: 2ϵ 1 ⊤ q ◦ > 2ϵ 1 ⊤ (B ◦ f ′ )− ϵ 2 1 ⊤ 1+ 2Cov{B ◦ f,q ◦ }− Var{B ◦ f} − 2Cov{B ◦ f ′ ,q ◦ }− Var{B ◦ f ′ } . LetusexploitLem.3.5inthesecondandthirdlinesandapplytheidentitiesVar{a ◦ +ϵ 1}= Var{a ◦ }andCov{a ◦ +ϵ 1,q ◦ − B ◦ f ′ }=Cov{a ◦ ,q ◦ − B ◦ f ′ }toobtain 2ϵ 1 ⊤ q ◦ > 2ϵ 1 ⊤ (B ◦ f ′ )+2Cov{a ◦ ,B ◦ f ′ }− 2Cov{a ◦ ,B ◦ f}− ϵ 2 1 ⊤ 1. Since f produced by the generic routing policy is independent of arrival statistics, we get Cov{a ◦ ,B ◦ f}=0. Replacing1 ⊤ q ◦ bytheQ expressionasdefinedin (3.2),wethenobtain 2ϵ Q > 2ϵ 1 ⊤ (B ◦ f ′ )+2Cov{a ◦ ,B ◦ f ′ }− ϵ 2 1 ⊤ 1. (3.36) Consider this time HD policy at β = 0 with the timeslot quantities ofq ⋆ ◦ (n) andf ⋆ (n). Let againf ′ (n) be produced by the fictitious algorithm at each slot n to stabilize the arrival a ◦ +ϵ 1,butthistime,givenq ⋆ ◦ (n). InlightofLem.3.4,G(f ′ ,q ⋆ ◦ ,n)6G(f ⋆ ,q ⋆ ◦ ,n)ateachslot 3.7ClassicalvsCombinatorialHeatProcess 63 n. Performing the similar steps of taking expectation, exploitingE{B ◦ f ′ } =E{B ◦ f ⋆ }+ϵ , translatingtheresultsintotheexpectedtimeaverageform,usingthefactthatCov{B ◦ f ⋆ ,a ◦ }= 0 asf ⋆ is independent of arrival statistics, and applying Lem. 3.5 by knowing that HD policy is throughputoptimalandsostabilizesa ◦ ,weobtain 2ϵ Q ⋆ 6 2ϵ 1 ⊤ (B ◦ f ′ )+2Cov{a ◦ ,B ◦ f ′ }− ϵ 2 1 ⊤ 1. (3.37) Comparing(3.36)and (3.37)alongwithϵ> 0leadtoQ ⋆ 6Q. Thismeanstheaveragenetwork delayunderHDwithβ =0remainslessthanorequaltothatunderanyotherD-classrouting policy, towhichwereferredas“generic”here. 3.7 ClassicalvsCombinatorial Heat Process Toformulateheatdiffusionongraph,weusethetheoryof combinatorialgeometryintroduced inCh.2,wherethenotionofchains-cochainsonacombinatorialdomainprovidesagenuine counterpartfordifferentialforms inclassicalgeometry. 3.7.1 Heatequationsonmanifolds OnasmoothmanifoldMchartedinlocalcoordinatesz,considerQ(z,t)asspatialdistribution oftemperature,F(z,t) asheatflux, and A(z,t) asscalarfieldofheatsources(withminus for sinks). Thelawofheatconservationentails ∂Q(z,t) ∂t =− divF(z,t)+A(z,t). (3.38) Fick’slaw relates the diffusive flux totheconcentration, postulatingthattheheatfluxgoes from warm regions of high concentration to cold regions of low concentration, with a magnitude that isproportionaltotheconcentrationgradient: F(z,t)=− σ (z)∇Q(z,t) (3.39) where σ (z) is thermal diffusivity that quantifies how fast heat moves through the material. Combining(3.38)and (3.39)together,weobtain ∂Q(z,t) ∂t =div σ (z)∇Q(z,t) +A(z,t). (3.40) 64 Heat-Diffusion: ParetoOptimalDynamicRouting To have a unique solution for this differential equation, besides time initial condition, one must prescribeQ conditionsonaspatial boundary∂M. 3.7.2 Heatequationsonundirectedgraphs In the context of combinatorial geometry, let us view a graph as a simplicial 1-complex and transfer elements of classical heat equations to this cell complex. In doing so, the smooth manifoldM is replaced by a 0-chain vector representing the discrete domain, the pointwise functionsQ(z,t)andA(z,t)arerespectivelyreplacedby0-cochainvectorsq(t)anda(t)(node variables),thelineintegralF(z,t)isreplacedby1-cochainvectorf(t)(edgevariable),and thethermaldiffusivity σ isreplacedbyavectorofedgeweightsσ . Recall from Ch. 2.2.6 that as a 1-complex, the graph structure is fully described by its node-edge incidencematrixB. (Theincidencematrix definedinSec.3.4 foradirected graph has the same structure except that the edge directions are substituted for the arbitrarily assigned algebraic topological edge orientations here.) Then on an undirected graph with a noded as the heatsink, combinatorialanalogueoftheclassical heatequations(3.38)–(3.40) areobtainedas ˙ q(t)=− Bf(t)+a(t), q d (t)=0 (3.41) f(t)=diag(σ )B ⊤ q(t) (3.42) ˙ q(t)=− Bdiag(σ )B ⊤ q(t)+a(t), q d (t)=0. (3.43) Notethattheboundary∂M onthemanifoldcollapsestothesinglenoded onthegraphatthe fixedzerotemperature,whichabsorbsalltheheatgeneratedbytheheatsources a(t). Enforcing boundary conditionq d (t)=0, one can eliminate the sinkd from (3.41)–(3.43), whichyieldsthereducedsetofcontinuous-timegraphheatequationsas f(t)=diag(σ )B ◦ ⊤ q ◦ (t) (3.44) ˙ q ◦ (t)=− L ◦ q ◦ (t)+a ◦ (t), L ◦ :=B ◦ diag(σ )B ◦ ⊤ . (3.45) whereasbefore,subscript◦ denotesareducedvectorormatrixthatdiscardstheentriescorre- sponding to the destination noded. The linear operatorL ◦ is called the Dirichlet Laplacian withrespecttothenoded,whichisasymmetricanddiagonallydominantmatrix. Further,as showninSec.2.3,foranyconnectedgraph,L ◦ is positivedefinite. 3.8WirelessNetworkThermodynamics 65 3.7.3 Heatequationsondirectedgraphs On a directed graph, the combinatorial heat conservation (3.41) remains unchanged, but the Fick’s law (3.42) must be modified to allow flow in only one direction. Let arbitrarily assigned edgeorientationsconcur withedge directions. Likethe undirectedcase,onecan dropthe sink noded from equations by fixing q d (t)=0 as boundarycondition. Thenweget thereduced set ofcontinuous-timeheatequationsonanuncapacitateddirectedgraphas f(t)=diag(σ )max 0, B ◦ ⊤ q ◦ (t) (3.46) ˙ q ◦ (t)=− ⃗ L ◦ q ◦ (t)+a ◦ (t), ⃗ L ◦ :=B ◦ diag(σ )diag II B◦ ⊤ q ◦ (t)≻ 0 B ◦ ⊤ . (3.47) We refer to ⃗ L ◦ as nonlinear Dirichlet Laplacian that acts on a directed graph and, unlikeL ◦ , is anoperand-dependentoperatorthatretainsneitherlinearitynorsymmetry. Remark 3.4. Forthe firsttime, heatdiffusion ondirected graphsisformulated viaa nonlinear Laplacian. This is in agreement with the recent work in [73] showing that heat diffusion on Finsler manifolds, the natural counterparts of directed graphs in continuous domain, leads to a nonlinear Laplacian. In the graph literature, different linear Laplacians have been proposed for directed graphs (see [74, Sec. 3] for a review). While successful to address some purely graphicalissues,theyarenotabletoconveythephysicsofthediffusionprocess,northeintrinsic nonlinearityduetotheone-wayflowrestrictions. Given finite heat sources, heat equations on a connected undirected graph always lead to finite temperatures at the nodes. However, for (3.47) to have a finite solution, each nonzero heat sourceneedstoconnecttothesink throughat least onedirectedpath. Ifthis basiccondition doesnothold,thenetworkflowproblemhasindeednosolutioninthesensethatthereisnoway totransferallcommodities,whichisheatinourcase,tothedestination. Definition 3.10. A nonzero heat source is feasible if it connects to sink through at least one directedpath,withthepathbeingdirectedfrom sourcetosinkforapositiveheat sourceand from sink to source for a negative heat source. A vector of heat sources is feasible if each of its nonzerocomponentsisfeasible. 3.8 WirelessNetwork Thermodynamics Thoughdefinedonadirectedgraph,theheatequations (3.46)–(3.47)stillrepresentadeterminis- tic,continuous-timeprocesswithnolinkinterference. Thelatter,particularly,makesthewireless 66 Heat-Diffusion: ParetoOptimalDynamicRouting problemquiteintractable. Nonetheless,thissectionadvocatesagenuinediffusionprocesson stochastic slotted-time interference networks by showing that under HD routing policy, the long-term average dynamics of an interference wireless network comply with non-interference combinatorialheatequationsonasuitably-weighteddirectedgraph. 3.8.1 ParetooptimalHDfluidlimit Fluid limit of a stochastic process is the limiting dynamics obtained by scaling in time and amplitude. Under very mild conditions,it isshownthatthesescaledtrajectoriesconvergetoa set of deterministic equations called fluid model. Using such a deterministic model, one can analyze rate-level, rather than packet-level, behavior of theoriginal stochastic process. Details arefound in[75,76]andreferencestherein. Fluidlimit: LetX(ω,t)bearealizationofacontinuous-timestochasticprocessX along a sample pathω. Define the scaled process X r (ω,t) := X(ω,rt)/r for anyr > 0. A deter- ministicfunction ˜ X(t) isafluidlimit ifthereexistasequencer andasamplepathω suchthat lim r→∞ X r (ω,t)→ ˜ X(t) uniformly on compact sets. For a stable flow network, the existence of fluid limits is guaranteed if exogenous arrivals are of finite variance. It is further shown that eachfluidlimit isLipschitz-continuous,andsodifferentiable, almosteverywherewithrespect toLebesguemeasureon[0,∞). Cumulativeprocess: Notethatthefluidtheoremisdefinedforcontinuous-timestochastic processes, while a wireless network is a slotted-time process. To resolve this issue, we derive a first-ordercontinuous-timeapproximationofwirelessnetworkdynamicsusingitscumulative model. Leta tot ◦ (n) andf tot (n) be respectivelythe vector ofcumulative node arrivals andlink transmissions up to slot n. In light of dynamic equation (3.10) and by assuming the initial conditionsa tot ◦ (0)=0 andf tot (0)=0,weobtain q ◦ (n)=q ◦ (0)+a tot ◦ (n)− B ◦ f tot (n). (3.48) Let ̂ f ij (n) be the predicted number of packets that linkij would transmit if it were activated at slotnandcomposethevector ̂ f(n). AlsoletT π (n)bethecumulativenumberoftimeslots,until slotn,in whichthescheduling vectorπ ∈Π hasbeen selected. Assumingtheinitialcondition T π (0)=0,onecanverifythat f tot (n)= X π ∈Π n X k=1 T π (k)− T π (k− 1) π ⊙ ̂ f(k) . (3.49) 3.8WirelessNetworkThermodynamics 67 The first parenthesis equals 1 if the scheduling vector π has been selected at slot k, and 0 otherwise. The term(π ⊙ ̂ f(k)) represents the number of packets that could be transmitted over each link if the scheduling vectorπ were selected. Note that a routing policy needs to determineeachentryof ̂ f(k) andselectaschedulingvectorπ ∈Π ateachtimeslot. General fluid equations: Given a sample path ω, we extend a slotted-time process to be continuous-time via linear interpolation in each timeslot interval(n,n+1). With no loss of generality,letexogenousarrivalsoccuratthebeginningofeachtimeslotsothata tot ◦ (t)represents cumulative arrivals by time t. Assuming normalized timeslots with the period of time unit, (3.48)directlyprovidesthefirstsetofstochasticgeneralfluidequationsas q ◦ (t)=q ◦ (0)+a tot ◦ (t)− B ◦ f tot (t) (3.50) a tot ◦ (t)=a ◦ t (3.51) witha ◦ being the time average expectation of the random arrivalsa ◦ (n). The second set of generalfluidequationsareobtainedfromthetimederivativeof (3.49)as ˙ f tot (t)= X π ∈Π ˙ T π (t) π ⊙ ̂ f(t) (3.52) ˙ T π (t) π ∈Π =    1 if π ischosenattime t 0 otherwise (3.53) X π ∈Π T π (t)=t with T π (t) nondecreasing. (3.54) Notethat (3.52)entailstheexistenceofaδ > 0 suchthat f tot ij (t ′ )− f tot ij (t)= X π ∈Π π ij ̂ f ij (t) T π (t ′ )− T π (t) foranyt ′ ∈[t,t+δ ]. Thisstatesthefactthatifalinkhasapositiveflowofpacketsattime t,the number of packets transmitted by the link in an interval[t,t ′ ]⊂ [t,t+δ ] is equal to the amount of time the link has been activated during[t,t ′ ] multiplied by its transmission rate prediction at timet, i.e.,alinearinterpolationontheclosedinterval[t,t ′ ]. Particular fluid equations: While (3.50)–(3.54) hold for any stable network operating under anarbitrarynon-idlingcontrolpolicy,eachpolicydetermines ̂ f(t)andT π (t)initsownparticular way. Referringto(3.6),uniclassParetooptimalHDpolicyenforces ̂ f(t) HD = min Φ B ◦ ⊤ q ◦ (t) + ,µ . (3.55) 68 Heat-Diffusion: ParetoOptimalDynamicRouting whereΦ represents the time average expectation ofΦ (n) as defined in (3.11). Note that the existence ofµ andρ is secured by (3.15) and (3.16). Referring now to(3.7) and (3.8), uniclass Pareto optimal HD policy determines the scheduling vector π (t) by solving the following max-weightoptimizationproblem: π (t)=argmax π ∈Π π ⊤ w(t) (3.56) w(t) HD = ̂ f(t)⊙ 2Φ B ◦ ⊤ q ◦ (t)− ̂ f(t) (3.57) wherew(t) isthevectorofweightsassignedbyHDpolicytoeachlinkattimet. Foracomparison,observethattheoriginalBPsolvesthesamemax-weightoptimization problem (3.56)tofindaschedulingvector π (t),butitenforces ̂ f(t) andw(t) tobe ̂ f ij (t) BP =    min{q i (t),µ ij } if q ij (t)>0 0 otherwise (3.58) w(t) BP = µ ⊙ B ◦ ⊤ q ◦ (t) + . (3.59) Theorem 3.4. (HDFluidModel) On a uniclass wireless network stabilized by Pareto optimal HDpolicy,everyfluidlimit ˜ X(t)= ˜ q ◦ (t), ˜ f tot (t), ˜ T π (t) satisfies HDfluidmodel, whichis definedasthecollectionofdeterministiccontinuous-timeequations(3.50)–(3.57). Proof. The proof follows the exact same line of argument proposed in [75, Theorem 2.3.2] and [76,Proposition4.12]. Remark 3.5. Itisimportanttodiscriminatebetweenfluidlimitandfluidmodelofadiscrete- time stochastic process. The former is the scaled process of the first-order continuous-time approximation for an arbitrary realization of the stochastic process, while the latter is a (set of)fullydeterministic,continuous-timeequation(s). Considernowawirelessnetworkunder HD routing policy, where the discrete-time stochastic processesq ◦ (n),f tot (n) andT π (n) have respectively the continuous-time fluid limits ˜ q ◦ (t), ˜ f tot (t) and ˜ T π (t). Then Th. 3.4 states that for large enough scaling factors, the fluid limit of every realization converges to a set of deterministic, continuous-time functionsq ◦ (t), f tot (t) and T π (t) which solve the HD fluid modelequations(3.50)–(3.57). 3.8WirelessNetworkThermodynamics 69 3.8.2 Thermodynamic-likepacketrouting ConsiderauniclasswirelessnetworkwithpacketsbeingroutedunderParetooptimalHDpolicy (microscopic flow). At each timeslot, HD policy activates a particular set of links to transmit a specific number of packets over them. Obviously, each link transmits packets at some slots and is switched off at some other slots. Letus now lookat the limit flowon each link, defined as the totalnumberofpacketstransmittedoverthelinkduringalargeperiodoftimedividedbythe time duration. We claim that observing average packet flow in limit (macroscopic flow), it takes theformofheatflowontheunderlyingdirectedgraphwithsuitably-weightededges. Considerathermalgraphwiththesamenode-edgeincidencematrixB ◦ andtheedgethermal diffusivity σ ij = φ ij . Associate with each arrivala i (n) on the wireless network a static heat source of intensitya i on the graph and fix zero temperature at the destination node. The flow of heatonthisdirectedgraphisgovernedby (3.46)–(3.47),whichprovidesthewirelessnetwork withastatic referencethermalmodelas f opt =Φ max 0, B ◦ ⊤ q opt ◦ (3.60) a ◦ = ⃗ L opt ◦ q opt ◦ , ⃗ L opt ◦ :=B ◦ Φ diag II B◦ ⊤ q opt ◦ ≻ 0 B ◦ ⊤ . (3.61) Notethatφ ij dependsnotonlyonthelinkcostfactorρ ij ,butalsoonthepenaltyfactorβ ,where varyingβ leadstodifferentedgeweightsandsotodifferentgraphtopologies. Recall that T π (t) represents the cumulative time until t in which the scheduling vector π ∈ Π has been selected. Obviously, each scheduling policy leads to its own specific T π (t). For instance, under Pareto optimal HD policy, T π (t) is determined by the HD scheduling (3.56)–(3.57),whiletheoriginalBPdeterminesitaccordingto(3.58)–(3.59). Definition 3.11. Underasequenceofwirelesslinkscheduling,theeffectivecapacity oneach linkisthetimeaverageexpectationofcapacitymadegenuinelyavailableonthatlink,viz., µ eff :=limsup τ →∞ 1 τ X π ∈Π τ X n=0 T π (n)− T π (n− 1) π ⊙ E{µ (n)} whereµ eff denotesthevectorofeffectivelinkcapacities. Observethattheclassicalheatequations(3.38)–(3.40),andtheircombinatorialcounterparts (3.44)–(3.45), take no limit in either flow direction or flow capacity. Then note that while (3.46)–(3.47) extend heat equations to directed graphs, they still consider no capacity limits 70 Heat-Diffusion: ParetoOptimalDynamicRouting onbranches. Infact,theunderneathassumptionisthattheflowofheatoneachdirectededge followstheFick’slawofdiffusion,notintervenedbytheedgecapacity. Assumption 3.2. Given an arrival ratea ◦ , there exists at least one sequence of wireless link scheduling under which the effective linkcapacitiesmeet therequirementofreferenceheatflow (3.60),whichisstatedbyf opt 4µ eff . Whileµ eff isanetworkcharacteristicandindependentofarrivals,satisfactionofAssum.3.2 does dependonarrivals. Further,foragivenarrivalrate, therecouldbealargenumberoflink scheduling sequencesthatmeettherequirement. Theorem 3.5. (Wireless Network Thermodynamics) Consider a uniclass wireless network that meets Assum. 3.1 and 3.2 under a stabilizable arrival rate. Then the HD fluid model (3.50)–(3.57)asymptoticallyconvergestothethermalmodel (3.60)–(3.61). Inparticular,HD fluidmodelwith β =0complieswithheatequationsonanunweighteddirectedgraph,andwith β =1 tothoseonaweighteddirectedgraphwithσ ij =1/ρ ij . Proof. Letq ⋆ ◦ (t) andf ⋆ (t) denote the HD fluid model variables. Consider the continuous-time Lyapunov function Y(t):= q ⋆ ◦ (t)− q opt ◦ ⊤ M ◦ q ⋆ ◦ (t)− q opt ◦ whereM ◦ = B ◦ B ◦ ⊤ − 1 B ◦ Φ B ◦ ⊤ representsthetimeaverageexpectationofmatrixM ◦ (n)as definedin (3.18). Takingtimederivativefrom Y(t),weobtain ˙ Y(t)= ˙ q ⋆ ◦ (t) ⊤ M ◦ ⊤ +M ◦ q ⋆ ◦ (t)− q opt ◦ . ExploitingLem.3.3inthelatterleadsto ˙ Y(t)6η ˙ q ⋆ ◦ (t) ⊤ M ◦ q ⋆ ◦ (t)− q opt ◦ (3.62) for an16η 63. As a positive coefficient, η has no impact on the Lyapunov argument and can simplybeomitted,butforthesakeofconsistencyweprefertokeepitinhere. To find an appropriate expression for ˙ q ⋆ ◦ (t), let us begin by plugging (3.51) in (3.50) and takingtimederivativetoobtain ˙ q ⋆ ◦ (t)=a ◦ − B ◦ ˙ f ⋆tot (t). (3.63) Notein (3.55)thattheentryof ̂ f(t)correspondingtolinkij specifiesthenumberofpackets the link will send per unit time if it is activated at timet. Thenf(t) identifies the vector of rate 3.8WirelessNetworkThermodynamics 71 of actual transmissions realized at timet. Assume now that the entry off(t) corresponding to linkij attimet isequal tox>0, i.e.,at timet thelink transmitsx number ofpacketsper unit time. Thenitshouldbeobviousthatthesameentryof ˙ f tot (t)attimetmustalsobeequalto x. In light oflim δ →0 f tot (t+δ )=f tot (t)+δ f(t), this can be explained more formally by the classicaldefinitionoflimitas ˙ f tot (t)= lim δ →0 f tot (t+δ )− f tot (t) δ =f(t). Further, (3.60)–(3.61) implya ◦ = ⃗ L opt ◦ q opt ◦ = B ◦ f opt . Exploiting these latter identities in (3.63)yields ˙ q ⋆ ◦ (t)=B ◦ f opt − B ◦ f ⋆ (t).ReturningtotheLyapunovargument,letussubstitute thelatterin (3.62)andutilizeequality (3.23)inLem.3.2toobtain η − 1 ˙ Y(t)6 f opt − f ⋆ (t) ⊤ Φ B ◦ ⊤ q ⋆ ◦ (t)− q opt ◦ . Multiplying both sides by two, adding and subtracting the termf ⋆ (t) ⊤ f ⋆ (t)+f opt⊤ f opt on the right-handside,andrecastingthe termsleadto 2η − 1 ˙ Y(t)6− 2f ⋆ (t) ⊤ Φ B ◦ ⊤ q ⋆ ◦ (t)− f ⋆ (t) ⊤ f ⋆ (t) (3.64a) + 2f opt⊤ Φ B ◦ ⊤ q ⋆ ◦ (t)− f opt⊤ f opt (3.64b) − 2f opt⊤ Φ B ◦ ⊤ q opt ◦ − f opt⊤ f opt (3.64c) + 2f ⋆ (t) ⊤ Φ B ◦ ⊤ q opt ◦ − f ⋆ (t) ⊤ f ⋆ (t) . (3.64d) Characterizing (3.64a) and (3.64b) on the wireless network givenq ⋆ ◦ (t), they respectively read − D(f ⋆ ,q ⋆ ◦ ,t)andD(f opt ,q ⋆ ◦ ,t). UnderAssum.3.1andinlightoftheHDfluidequations (3.55) and(3.57),theimmediateresultofTh.3.4isthatateachtimet,HDfluidlimitmaximizesthe D functionalcomparedtoanyalternativeforwardingthatsatisfieswirelessnetworkconstraints. The f opt obviously meets the directionality constraints due to the structure of the reference thermal model. It also meets the capacity constraints due to Assum. 3.2. Hence, (3.64a)+(3.64b)60 whichsimplifiestheLyapunovinequalityto 2η − 1 ˙ Y(t)6− 2f opt⊤ Φ B ◦ ⊤ q opt ◦ − f opt⊤ f opt (3.65a) + 2f ⋆ (t) ⊤ Φ B ◦ ⊤ q opt ◦ − f ⋆ (t) ⊤ f ⋆ (t) . (3.65b) 72 Heat-Diffusion: ParetoOptimalDynamicRouting Wenowcharacterize (3.65a)and (3.65b)onthereferencethermalmodelandlet H(f):=2f ⊤ Φ B ◦ ⊤ q opt ◦ − f ⊤ f. It canbe shownthat givenq opt ◦ , themaximum ofH occurs atf =f opt produced by heat flow. Toseethis,letusrephraseH as H(f)= X ij∈E 2φ ij q opt ij f ij − f ij 2 wheredirectionalityconstraintsentailf ij >0. TomaximizeH,onethenneedstoassignf ij =0 ifq opt ij 6 0, andf ij = φ ij q opt ij otherwise. Putting this back in a matrix form, we arrive at the same expression asf opt in (3.60). Further, the maximizingf is unique by the reason that a givenq opt ◦ leadstoauniqueB ◦ ⊤ q opt ◦ ,andsotouniqueq opt ij components,asthematrixB ◦ has fullrowrank. FromH(f opt )>H(f ⋆ (t)),wethenobtain(3.65a)+(3.65b)60,whichinlight of16η 63 leadsto ˙ Y(t)60. LetΩ bethelargestinvariantsetinthesetofallq ⋆ ◦ (t)trajectoriesforwhich ˙ Y(t)=0. Since Y(t) is a non-negative and radially unbounded function with ˙ Y(t)6 0, LaSalle’s invariance principlestatesthateverytrajectoryq ⋆ ◦ (t)asymptoticallyconvergestoΩ . Itremainstoshowthat Ω contains only the trivial trajectory ofq ⋆ ◦ =q opt ◦ . If ˙ Y = 0, then (3.65) entailsH(f ⋆ (t)) = H(f opt ). Further,wepreviouslyshowedthatf opt maximizesH andisunique,whichimplies f ⋆ =f opt . (3.66) Theintentionallydroppedtimevariable(t)in(3.66)emphasizesthatf ⋆ (t)turnstobestationary byconvergingtof opt ,whichinturn entailsq ⋆ (t) beingconvergedtoastationaryq ⋆ too. Givenq ⋆ , the equality (3.66) entails thatf opt must maximizeD, which impliesf opt ij = 0 if q ⋆ ij 6 0, and f opt ij = φ ij q opt ij otherwise. In a matrix form, this is equivalent to f opt = Φ max 0, B ◦ ⊤ q ⋆ ◦ . Puttingthelatteragainst (3.60)leadsto max 0, B ◦ ⊤ q ⋆ ◦ =max 0, B ◦ ⊤ q opt ◦ . (3.67) Consider a directed edge ad with its head at the destination node, which has zero queue on thewirelessnetworkandzerotemperatureonthereferencethermalmodel. By (3.67),q ⋆ a and q opt a must be equal. Repeating this argument eventually yields(q ⋆ ◦ ) + = (q opt ◦ ) + , as any node with positive queue (resp. positive temperature) on the wireless network (resp. on the reference 3.9HDMinimumRoutingCostatβ =1 73 thermalmodel)mustbeconnectedtothedestinationnodedthroughadirectedpath. Further, observe that q ⋆ ◦ < 0 as queues cannot be negative in a wireless network, and q opt ◦ < 0 as temperatures cannot fall below zero in a thermal system with no negative heat source. Thus, q ⋆ ◦ =q opt ◦ ,whichtogetherwith (3.66)concludetheproof. Remark3.6. Assump.3.2examinesifitispossibleinprincipletostabilizethewirelessnetwork such thatitsfluidlimitfollows uncapacitatedheatequations. Wefullyrevokethisassumption in Ch. 5 by developing diffusion equations on capacitated directed graphs and showing that the fluidequations (3.50)–(3.57)stillrespectthemwithnoneedofsatisfyingAssum.3.2. Infact, wesolveamorecomplicateddiffusionprobleminCh.5,wherenotonlydirectededgeshave limitedcapacity,butflowswithdifferentdestinationsneedtobecarriedoverthenetwork,which raisesthechallengeofoptimaldesignationofedgecapacitiestoeachofthem. 3.9 HDMinimumRouting Cost atβ = 1 To establish the second pillar of HD Pareto optimality, this section shows, via Dirichlet’s principle, that average quadratic routing cost is minimized under HD policy withβ =1. In fact, we show a more general result that Pareto optimal HD with anyβ ∈[0,1] solves the following β -dependentoptimizationproblem: Minimize X ij∈E (f ij ) 2 /φ ij (3.68) whereβ =1 leads toφ ij = 1/ρ ij , which recovers (3.3) on minimizing the average quadratic routingcostR. 3.9.1 ClassicalDirichletprinciple Considertheclassicalheatdiffusionequations (3.38)–(3.40)subjecttoconstantheatsources A(z). In a steady-state thermal conduction, the law of heat conservation entails that the amount of heat entering any region of manifoldM is equalto the amount of heatleaving outthe region. Thus, while partial derivatives of temperature with respect to space may have either zero or nonzerovalues,alltimederivativesoftemperatureatanypointonMwillremainuniformly zero. ThisleadstotheclassicalPoissonequation div σ (z)∇Q(z) +A(z)=0 74 Heat-Diffusion: ParetoOptimalDynamicRouting which formulates stationary heat transfer by substituting zero for the time derivative of tempera- turein (3.40). Dirichlet’sprinciplethenstatesthatthePoissonequationhasauniquesolution thatminimizestheDirichletenergy E D Q(z) := Z M 1 2 σ (z)∥∇Q(z)∥ 2 − Q(z)A(z) dz amongalltwicedifferentiablefunctions Q(z) thatrespecttheboundaryconditionson∂M. 3.9.2 CombinatorialDirichletprinciple Toderivethecombinatorialanalogue ofPoissonequationonundirectedgraphs,oneidentifies theclassicaldiv withtheboundaryoperatorB andtheclassicalgradient∇withtheminusof coboundaryoperatorB ⊤ . Fixingq d (t)=0 yields − L ◦ q ◦ +a ◦ =0 (3.69) whichcorrectlyrealizes (3.45)insteady-state. Notethat theequation hasno timevariable(t), sinceitrepresentsthesteady-statecondition. Remark 3.7. In vector calculus, the gradient of a scalar field is positive in the direction of field increase. On a graph, however, wetake the gradient of a nodevariable positive in the direction of decrease of the variable. By the same reason, the classical Laplace operator is a negative semi-definiteoperator,whilethegraphLaplacianisapositivesemi-definitematrix. It is not difficult to see that, like the classical case, the equation (3.69) has a unique solution thatminimizesthecombinatorialDirichletenergy E D (q ◦ ):= 1 2 q ◦ ⊤ L ◦ q ◦ − q ◦ ⊤ a ◦ . (3.70) TheproofofDirichlet’sprincipleismuchsimplerinthecombinatorialcase. Infact,asL ◦ is positive definite, E D (q ◦ ) is strictly convex and so has a minimum at the critical point, where its first order variation vanishes, which readily leads to the combinatorial Poisson equation (3.69). 3.9.3 NonlinearDirichletprinciple Essentially,thePoissonequationonadirectedgraphshouldcapturethesteady-statebehaviorof combinatorial nonlinear diffusion process (3.47) subject to constant heat sourcesa ◦ . This leads 3.9HDMinimumRoutingCostatβ =1 75 tothefollowingnonlinearPoissonequation: − ⃗ L ◦ q ◦ +a ◦ =0. (3.71) Observethatin (3.70),thedirectionalderivativeof 1 2 q ◦ ⊤ L ◦ q ◦ alongq ◦ issimplyL ◦ q ◦ ,as appeared in (3.69), by the reason thatL ◦ is a symmetric positive definite matrix. On a directed graph,however,difficultyarisesfromthefactthatcontrarytolinearLaplacian L ◦ onundirected graphsthatisasymmetricpositivedefinitematrix, ⃗ L ◦ isanoperand-dependentoperatorthat retainsneitherlinearitynorsymmetricity. Thus,theeasywayofprovingDirichlet’sprincipleon undirectedgraphsceasestoexisthere,asonecannotclaimthat ⃗ L ◦ q ◦ in(3.71)isthedirectional derivativeof 1 2 q ◦ ⊤ ⃗ L ◦ q ◦ . Nonetheless,weextendtheconceptofcombinatorialDirichletprinciple todirectedgraphsbythenexttheorem. Theorem 3.6. (NonlinearDirichletPrinciple)Givenafeasibleheatsourcea ◦ onadirected graph, the nonlinear Poisson equation (3.71) has a unique solution that minimizes the following functional,namedasnonlinearDirichletenergy: ⃗ E D (q ◦ ):= 1 2 q ◦ ⊤ ⃗ L ◦ q ◦ − q ◦ ⊤ a ◦ . (3.72) Proof. Onecanverify,usingthe ⃗ L ◦ structurein (3.47),that ⃗ E D (q ◦ )= 1 2 q ◦ ⊤ B ◦ + diag(σ ) B ◦ ⊤ q ◦ + − q ◦ ⊤ a ◦ where each entry ofB ◦ ⊤ q ◦ represents temperature-difference along the corresponding edge. Letq ∗ ◦ be the ⃗ E D minimizing solution and let us rearrange and partitionB ◦ ⊤ q ∗ ◦ into positive, zeroandnegativecomponents. Accordingly,B ◦ getspartitionedintoB ⊕ ,B ∅ andB ⊖ ,which respectively contain the incidence information of edges with positive, zero and negative values inB ◦ ⊤ q ∗ ◦ . Likewise,σ getspartitionedintoσ ⊕ ,σ ∅ andσ ⊖ . Thenatq ◦ =q ∗ ◦ ,weobtain ⃗ E D (q ∗ ◦ )=− q ∗⊤ ◦ a ◦ + 1 2 q ∗⊤ ◦ B ⊕ + diag(σ ⊕ ) B ⊤ ⊕ q ∗ ◦ + + 1 2 q ∗⊤ ◦ B ∅ + diag(σ ∅ ) B ⊤ ∅ q ∗ ◦ + (3.73a) + 1 2 q ∗⊤ ◦ B ⊖ + diag(σ ⊖ ) B ⊤ ⊖ q ∗ ◦ + . (3.73b) 76 Heat-Diffusion: ParetoOptimalDynamicRouting Observe that (3.73a) is strongly zero due to the (·) + operation. On the other hand, (3.73b) vanishessinceB ⊤ ∅ q ∗ ◦ =0. Inlightof(B ⊤ ⊕ q ∗ ◦ ) + =B ⊤ ⊕ q ∗ ◦ ,wethenobtain ⃗ E D (q ∗ ◦ )= 1 2 q ∗⊤ ◦ B ⊕ diag(σ ⊕ )B ⊤ ⊕ q ∗ ◦ − q ∗⊤ ◦ a ◦ . (3.74) Sincea ◦ is feasible, each nonzero heat source connects to the sink through at least one directed path. Thus,underanyflowthatkeeps q ◦ entrywisefinite,theedgeswithpositivetemperature- difference build a connected graph with the node d. On the other hand,q ∗ ◦ is entrywise finite as itminimizes ⃗ E D (q ◦ ),andsothecorrespondingedgesinB ⊕ buildaconnectedgraphwiththe noded. ThisimpliesthatB ⊕ diag(σ ⊕ )B ⊤ ⊕ isapositivedefinitematrix. Thus,thefunctional 1 2 q ◦ ⊤ B ⊕ diag(σ ⊕ )B ⊤ ⊕ q ◦ − q ◦ ⊤ a ◦ isstrictlyconvexinq ◦ andsofindsits minimumatthecriticalpoint,whereitsfirstordervariationwithrespectto q ◦ vanishes. Compar- ing this with (3.74), it turns out that the minimizingq ∗ ◦ must satisfya ◦ =B ⊕ diag(σ ⊕ )B ⊤ ⊕ q ∗ ◦ . UtilizingB ⊤ ⊕ q ∗ ◦ = (B ⊤ ⊕ q ∗ ◦ ) + and addingB ⊖ diag(σ ⊖ )(B ⊤ ⊖ q ∗ ◦ ) + +B ∅ diag(σ ∅ )(B ⊤ ∅ q ∗ ◦ ) + in lightof(B ⊤ ⊖ q ∗ ◦ ) + =(B ⊤ ∅ q ∗ ◦ ) + =0,onecanrephrasea ◦ =B ⊕ diag(σ ⊕ )B ⊤ ⊕ q ∗ ◦ as a ◦ =B ◦ diag(σ ) B ◦ ⊤ q ∗ ◦ + = ⃗ L ◦ q ∗ ◦ which recovers the nonlinear Poisson equation (3.71) atq ∗ ◦ . Further,q ∗ ◦ is unique as it needs to minimizethestrictlyconvexfunctional 1 2 q ◦ ⊤ B ⊕ diag(σ ⊕ )B ⊤ ⊕ q ◦ − q ◦ ⊤ a ◦ . Remark 3.8. Though Dirichlet’s principle on undirected graphs has been known for long time, its extension to directed graphs is completely new to literature. As a model of heat flow on directed graphs, one can conceptualize a resistive network with a diode added to each edge [77]. Electrical current—the counterpart of combinatorial heat flux—moves along negative gradient of voltage, but only under the condition of respecting the diode direction. Another example is a piping network of liquid/gas with a check valve on each line. Again, the liquid/gas flows along negativegradientofpressure,whileeachcheckvalveallowstheflowinonlyonedirection. 3.9.4 Quadraticroutingcostminimization The framework of Th. 3.6 is not yet aligned with what we need for the optimization prob- lem (3.68). Thenexttheoremresolvesthisincongruitybyshowingthatminimizingthefunc- tional (3.72) is indeed the dual of minimizing network energy dissipation, known as Thomson’s principle,onthedirectedgraphwith zerodualitygap. 3.9HDMinimumRoutingCostatβ =1 77 Theorem3.7. (NonlinearThomsonPrinciple)MinimizingthenonlinearDirichletenergy(3.72) subjecttothenonlinearPoissonequation (3.71)isequivalenttominimizingtotalenergydissi- pationonthegraphsubjecttoflow conservationatthenodes,statedby min f<0 ⃗ E R (f):=f ⊤ diag(σ ) − 1 f s.t. B ◦ f =a ◦ (3.75) wheref<0isimposedbynetworkdirectionality. Further,temperaturesatthenodesplaythe naturalroleoftheLagrangemultipliersinthedualoftheoptimizationproblem (3.75). Proof. ConsidertheThomsonprinciple (3.75)astheprimaloptimizationproblemandletus constructitsLagrangiandualproblemas max λ min f<0 L(λ ,f):=f ⊤ diag(σ ) − 1 f +2λ ⊤ a ◦ − B ◦ f whereλ is the vector of Lagrange multipliers. From the first order condition ∇ f L = 0, we getf opt =diag(σ )B ◦ ⊤ λ . Thenenforcingtheconstraintf opt <0leadstoB ◦ ⊤ λ <0,whichis equivalenttoB ◦ ⊤ λ =(B ◦ ⊤ λ ) + . Thus,weobtain f opt =diag(σ )(B ◦ ⊤ λ ) + . (3.76) Pluggingthisf opt intotheLagrangianL andutilizingthestructureof ⃗ L ◦ in (3.47),weobtain L(λ )=− λ ⊤ ⃗ L ◦ λ +2λ ⊤ a ◦ . Thenthedualproblemreadsmax λ L(λ ),whichisequivalentto min λ 1 2 λ ⊤ ⃗ L ◦ λ − λ ⊤ a ◦ . (3.77) Further, asf<0 makes a convex set andL(λ ,f) is a convex function, the duality gap is zero, andsoboththeprimalanddualproblemsresultinthesameoptimalsolution. Comparing(3.77)withthenonlinearDirichletequation (3.72),itremainstoshowthatthe Lagrangian multipliersλ are identical to the node temperaturesq ◦ . In (3.76), multiplying both sidesbyB ◦ andusingthe ⃗ L ◦ expression,weobtain ⃗ L ◦ λ =B ◦ f opt =a ◦ (3.78) 78 Heat-Diffusion: ParetoOptimalDynamicRouting where the second equality comes from the constraint in the primal problem (3.75). By Th. 3.6, on the other hand, the nonlinear Poisson equation ⃗ L ◦ q ◦ =a ◦ has a unique solution. Putting thisagainst (3.78)leadstoλ =q ◦ ,whichconcludestheproof. It is worth comparing the minimization problem (3.75) with the celebrated law of least energy dissipation on resistive networks. In essence, Th. 3.7 extends the law to directed graphs, or to nonlinear resistive-diode networks for that matter [77]. The upshot is then due to the connection between heat diffusion on capacitated directed graphs and HD fluid limit, which bringstogethercircuittheoryandwirelessnetworkingunderoneumbrella. Theorem3.8. (HDMinimumRoutingCost)Considerawirelessnetworkunderastabilizable arrivalrate. UnderAssum.3.1and3.2,uniclassParetooptimalHDpolicysolvestheβ -dependent optimizationproblem(3.68). Inparticular,HDpolicywithβ =1minimizestheaveragequadratic routingcostR asdefinedin (3.3). Proof. It was shown by Th. 3.6 that if a ◦ is feasible, then under the nonlinear heat equa- tions (3.60)–(3.61), the stationary value of the nonlinear Dirichlet energy ⃗ E D (q ◦ ) is strictly minimized. ItwasshownbyTh.3.7,ontheotherhand,thatminimizing ⃗ E D (q ◦ )isequivalentto minimizing the stationary value of total energy dissipation ⃗ E R (f) on the graph. Then the proof immediatelyfollowsfromTh.3.5whichstatesthatunderastabilizablearrivalratea ◦ ,HDfluid model complieswith thenonlinear heatequations(3.60)–(3.61). Notethat ifa ◦ is stabilizable, i.e.,itsatisfiescondition (3.17),thenitsfeasibilityistrivialinthesenseofDef.3.10. In light of Th. 3.5, every expected time average value on a stochastic wireless network governed by HD policy follows the corresponding stationary value produced by nonlinear heat equationsonthesuitablyweightedunderlyingdirectedgraph. Inparticular,theβ -dependent objective function in (3.68) complies with the total energy dissipation ⃗ E R (f) on the graph weighted byσ ij =φ ij . By the same token, the average quadratic routing costR complies with ⃗ E R (f) onthegraphweightedbyσ ij =1/ρ ij . Remark 3.9. As Rem. 3.6 explained, Assum. 3.2 ensures that the link capacity constraints of wireless network do not intervene the Fick’s law on its underlying directed graph. Again, thisassumption isfullyrevoked in[77, 78]bydevelopingDirichlet’sprincipleoncapacitated directedgraphsandshowingthatHDfluidmodelstillcomplieswithit. 3.10ParetoOptimality 79 3.10 ParetoOptimality Minimizingaveragenetworkdelayandminimizingaverageroutingcostareoftenconflicting objectives,meaningthatasonedecreasestheotherhastoincrease. Thisnaturallyleadstoamulti- objectiveoptimizationframework. ThenthefavoriteoperatingpointslieontheParetoboundary that corresponds to equilibria from which any deviation results in performance degradation in at least one objective. In other words, a Pareto optimal solution is a state of allocation of resourcesfromwhichitisimpossibletoreallocatesoastomakeanyoneobjectivebetteroff withoutmakingatleastanotherobjectiveworseoff. 3.10.1 StrongParetooptimalityfornonuniformlinkcosts We have shown that HD withβ =0 minimizes the average network delayQ among allD-class routingpolicies—solvingtheoptimizationproblem (3.2). WehavealsoshownthatHDwith β =1strictlyminimizesthequadraticroutingcostRamongallstabilizingroutingalgorithms— solving the optimization problem (3.3). Consider now the region of operation built on joint variables(Q,R)inwhichQisachievablebyD-classroutingpolicies(possiblyunfeasible). The nexttheoremshowsthatHDpolicyoperatesontheParetoboundaryofthis(Q,R)regionby alteringβ ∈[0,1],solvingthemulti-objectiveoptimizationproblem (3.4). Theorem 3.9. (HD Pareto Optimality) Consider a uniclass wireless network that meets As- sum. 3.1 and 3.2 under a stabilizable arrival rate. Suppose that the operating region built on all possible joint variables(Q,R) withQ produced by aD-class routing policy is convex. then HDpolicyoperatesontheParetoboundaryof(Q,R) regionbyalteringβ ∈[0,1]. Proof. Astheregion(Q,R)hasaconvexParetoboundary,itisknownthattheentireboundary can be reached by the weighted-sum method [79], where the Pareto front is obtained by altering a weight between the two objective functions. Then the proof follows by observing that the HD policy minimizesQ atβ =0, minimizesR atβ =1, and varies the weight on these two objectivesbyalteringtheLagrangemultiplierβ between0and1. Itisworthtonotethatinthecaseofnon-convexParetoboundary,HDwithβ ∈[0,1]still covers the points on convex parts of the boundary, though some Pareto optimal points lie on non-convexparts[79]. Remark 3.10. To the best of our knowledge, this is the first time a network layer routing policy providesParetooptimalperformancewithrespecttoaveragenetworkdelayandroutingcost, withoutrequiringanyknowledgeoftrafficandtopology. 80 Heat-Diffusion: ParetoOptimalDynamicRouting 01 [ , ]   Feasible space of objectives V = 0 (Original BP)  V HD policy weak Pareto boundary weak Pareto boundary Average network delay Average routing cost Q R Fig.3.7 Graphical description of weak Pareto boundary with respect to average queue congestion andtheDirichletroutingcostwhenalllinkcostfactorsconvergetoone,contrastingtheperformance ofPareto optimal HD with V-parameterBP. 3.10.2 WeakParetooptimalityforunitlinkcosts Whenthecostfactorsinalllinksconvergetoone,wegetφ ij =1forallβ ,andsotheperformance of HD policy turns to be independent of the penalty factor β . Considering this observation alongwithTh.3.3impliesthattheaveragenetworkdelayQmustbeminimizedforallβ ∈[0,1]. Considering it along with Th. 3.8, on the other hand, implies that the average quadratic routing costR mustalsobeminimizedforallβ ∈[0,1]. Holdingthesetworequirementsatthesame timeentailsthatQandR mustbeminimizedtogether,whichequivalentlymeansthatthePareto boundary of(Q,R) region must shrink into one single point. Such an operating point is called weakly Pareto optimal in the sense that no tradeoff is allowed as it is impossible to strictly improveatleastoneoperatingobjective. Theupshotisformalizedbythenextcorollary. Corollary 3.1. Consider a uniclass wireless network under the same condition of Th. 3.9. Suppose the cost factors for all wireless links converge to one. Then the Pareto boundary of (Q,R) regionshrinkstoapointatwhichHDpolicyoperatesforallβ ∈[0,1]. Underunitcostfactorforalllinks,Fig.3.7providesagraphicalillustrationofthefeasible regionbuilton(Q,R). ItemphasizesHDoperationattheweaklyParetooptimalpointforall β ∈[0,1] incomparisonwiththeperformanceofV-parameterBPforV ∈[0,∞). 3.11Conclusion 81 3.11 Conclusion We have introduced a network layer routing policy, called Heat-Diffusion (HD), for uniclass wirelessnetworksthat(i)isthroughputoptimal,(ii)minimizesaveragequadraticroutingcost, (iii)minimizesaveragenetworkdelaywithinanimportantclassofroutingpolicies,(iv)provides a Pareto optimal tradeoff between average network delay and quadratic routing cost, and (v) en- joysthesamealgorithmicstructure,complexityandoverheadasBack-Pressure(BP)routing policy. Further,ParetooptimalHDpolicyisstronglyconnectedtotheworldofheatcalculus in mathematics, which we believe opens the door to a rich array of theoretical techniques to analyze and optimize wireless networking. For example, such a connection provides a new way ofanalyzingtheimpactofwirelessnetworktopologyonstabilityandcapacityregion[80]or on delay/routing energy performance [81]. A decentralized HD protocol has been pragmati- callyimplementedandexperimentallyevaluatedin[82]fordatacollectioninwirelesssensor networks, including a comparative analysis of its performance with respect to the Backpressure CollectionProtocol[56]. Thoughmotivatedbywirelessnetworks,theHDframeworkcanbeextendedinvariousways tootherapplicationareas. Among themispacketschedulinginhighspeedswitcheswithalot of attention in recent years. Resource allocation problems in manufacturing and transportation alsofallwithinthescopeofthemodelweconsideredhere. Chapter4 MulticlassMinimumDelayRouting We consider the problem of average network delay minimization on multiclass, multihop, stochasticwirelessnetworkssubjecttointer-channelinterferenceandtime-varyingtopology. Anetworkroutingcontrolisproposedtosolvethisproblemintheclassofallpolicieswhose control decisions are functions only of current queue congestion and current channel states, includingthosewithperfectknowledgeoftheprobabilitiesassociatedwithfuturerandomevents. To this end, we extend the results of minimum delay with Heat-Diffusion routing policy, which was developed for uniclass networks in Ch. 3, to multiclass routing problems. As important features of the proposed control policy, it is throughput-optimal in the sense that it can stabilize queuesforanystabilizabletrafficmatrix,itisrobusttovaryingnetworktopologyandarrival rates, and it is implemented without requiring any knowledge of statistics and probabilities of the system. Delay performance of the proposed control policy is analyzed via stochastic discrete-timeLyapunovtheoremandisevaluatedviasimulation. 4.1 Introduction Consideramulticlassslotted-timestochasticwirelessnetwork,wherethechannelconditions are time-varying according to some (unknown) probability laws, and where simultaneous transmission over two channels fail if they have interference. Packets of the same size randomly arrive at any node, while destining for different destinations and perhaps requiring multihop routingpaths. Ateachtimeslot,anetworkcontrollerobservesqueuecongestionandchannel conditionstomakeacontrolactionthatdetermineswhichsetofchannelsshouldbeactivated andhowmanypacketsfromeachclassshouldbetransmittedoverthem. Inthischapter,thegoal 84 MulticlassMinimumDelayRouting of the controller is to minimize the time average total queue congestion in the network, which is proportionaltoaveragenetworkdelaybyLittle’sTheorem[83]. Achieving this goal for a general case requires the Markov structure of topology process, plus arrival and channel state probabilities. Then in theory, the solution is obtained through dynamicprogrammingforeachpossibletopologyalongwithsolvingaMarkovdecisionproblem. Byevenhavingalltheserequiredinformation,stillthenumberofqueuebuffersandchannel states increase exponentially with the size of network, which make dynamic programming and Markov decision theory impractical. In fact, even for the case of a single channel, it is difficult toimplementtheresultingstochasticalgorithms[70]. Whilehavingapracticalsolutionfora general case seems dubious, this paper solves the problem within an important class of network controllers, withoutrequiringanyof theabove-mentionedinformationandwithoutdealingwith dynamicprogramingorMarkovdecisionprocess. We consider the class of all dynamic network controllers that make routing decision as a purefunctionofcurrentqueuecongestionandcurrentchannelstates,includingtheoneswith perfect probability knowledge of arrivals and channel states. As an important criterion, this class allows designing throughput-optimal network controllers that can stabilize all arrival rates in the network capacity region. Specifically, this class includes the following two important groupsofnetworkcontrolpolicies: (Randomized) Allstationaryrandomizedalgorithmsthatmakeindependent,stationaryand randomized transmission decisions at each timeslot based only on current channel ca- pacities and so independent of both queue congestion and channel quality factors. While such controllersexist in theory[72, 63], theyare intractable inpractice as theytypically requireafullknowledgeofarrivalstatistics,channelstateprobabilitiesandtheMarkov structureofnetworktopologyprocess. Besidesthis,thenetworkcontrollerwouldstill needtosolveadynamicprogrammingproblemforeachtopologystate,wherethenumber ofstatesgrowsexponentiallywiththenumberofchannels. Whilethesecontrollersare notpracticallyattractive,thefactthattheytheoreticallyexistplaysacrucialroleinthe analysisandevaluationofwirelessnetworkproblems[72,63]. (Max-weight) All opportunistic max-weight algorithms that do not incorporate the Markov structure of network topology process into their decisions. These controllers make a transmission decision at every timeslot via locally weighting each link and then globally schedulingasetoflinkswithmaximumsumweight. Ourproposedcontrolpolicyhere, 4.1Introduction 85 also the well-known Back-Pressure (BP) control policy [44] and all of its follow-up derivations,belongtothisgroup. 4.1.1 Relatedworks The seminal paper on congestion-aware routing control by Tassiulas and Ephremides [44] showed that the link queue-differential, capacity-based BP is throughput-optimal under very general conditions. We analytically prove that our proposed control is also throughput-optimal underthesamegeneralconditionsandwiththesamecomplexityasBP,whilealsominimizes thetimeaveragetotalqueuecongestionintheclassofallnetworkcontrollersthatrelyonly on currentqueuecongestionandcurrentchannelstates. ThestochasticnetworkoptimizationframeworkforthedesignandanalysisofBPhasproven to be a very popular research domain, with many new theoretical results in recent years to furtherenhancetheoriginalframework,wheremanyofthemhavefocusedonenhancingthe BP delay performance. Shadow queues has enabled BP to handle multicast sessions with reduced number of actual queues that need to be maintained [52, 84, 85]. A delay-based BP formulation has shown that a significantly lower delay can be obtained by using the cumulative time packet age queue [53]. Incorporating last-input-first-output (LIFO) service into BP has offered a better delay quality [ 54]. Adaptive redundancy has been designed to improve the low-rate delay performance of BP in intermittently connected mobile networks [55]. Using graphembedding,[49]combinedBPwithgreedyroutinginhyperboliccoordinatestoobtain a throughput-delay tradeoff. The framework has also been extended to handle finite buffer sizes [59]. Other researchers have focused on making BP scheduling more distributed so that it can be implemented more easily [86,61,62]. More recently, there have been several reductions of BPtheorytopractice,in theformofpracticallyimplementedandexperimentallyevaluated distributedprotocols[87,57,56,88,89]. 4.1.2 Organization Afterthisintroduction,thenextsectionpresentsdynamicmodelofamulticlassqueuingnetwork followed by network capacity region and multiclass BP policy. We propose in Sec. 4.3 the minimum delay policy for multiclass networks with and without the assumption of transmitting fromoneclassperlinkpertimeslot. Thesubstantialdynamiccharacteristicofdelayminimizing controlisintroducedinSec.4.4. Forourproposedcontrolpolicy,weprovideanalyticalproofof throughput optimality in Sec. 4.5 and of minimum routing delay in Sec. 4.6. Simulation results 86 MulticlassMinimumDelayRouting in Sec. 4.7comparethe delayperformanceofourcontrolpolicywithmulticlassBP algorithm. ThechapterisconcludedinSec.4.8. 4.1.3 Notation WeusethesamenotationasinCh.3(seeSec.3.1.4). Todenotemulticlasshyper-arrays,weuse blackboard bold typeface, such asf f, where certain lines of the symbol are doubled. ForA as a matrix,vec(A) transformsittoacolumnvector. 4.2 Preliminaries The network is described by a simple, directed connectivity graph with set of nodesV and directededgesE. Newpacketsrandomlyarriveatdifferentnodesandhavedifferentdestinations in a setK⊆V . Packets of the same destination form a class. Each nodei∈V holds a separate queueq (d) i foreachd-classtotransmitoveritsoutgoinglinks. Weassumethatpacketsarenot sent to trapping nodes in the network, i.e., when a node acceptsd-class packets (ord-packets in short),itimplicitlysuggeststhatthereexistsatleastonepossibleroutefromthatnodetothe destinationd. While this assumption is not required for any of our analytical results, it ensures that a dynamic routing control, with no predefined routing path constraint, does not send a packettoatrappingnodethatpreventsitfromeverreachingitsdestination. Contrary to wireline networks where links are independent resources, in a wireless network twolinkscannotsimultaneouslytransmitiftheyhaveinterference. Defineascheduleasaset of links in which no two links interfere with each other and call it maximal if no more links can be added to it without violating the interference constraints. Each maximal schedule is representedasavector,referredtoasschedulingvector,inwhicheachentrytakesthevalue1 if the corresponding link is included in the maximal schedule, and 0 otherwise. For a given connectivity graph(V,E), we assume that each maximal scheduling vectorπ takes values in a finite schedulingsetΠ ,whichisthecollectionofallavailablemaximalschedules. Observethattheschedulingsetvariesaccordingtointerferencemodel. Ourresultsarevalid forthecategoryofallinterferencemodelsinwhichanodecannottransmitpacketstomorethan one neighbor at each timeslot, i.e., a node may receive packets from several of its incoming linksandatthesametimemaytransmitpacketsoveroneofitsoutgoinglinks. Tothebestof our knowledge, interference constraints in all current network layer protocols, including general K-hopinterferencemodels,fallinthiscategory. 4.2Preliminaries 87 Adiscrete-timestochasticprocessx(n) iscalledstableif x:=limsup τ →∞ 1/τ X τ − 1 n=0 E{x(n)}<∞ (4.1) whereE denotes expectation. This definition of stability is frequently called strong stability incontrastwithotherweakerstabilitydefinitionssuchasratestability. Itisshownthatstrong stabilityentailsalloftheotherformsofstability[63,Theorem2.8]. A queuing network is called stable if all its queues are stable. A traffic matrix, which represents the arrival rate of different classes into different nodes, is called stabilizable if there existsa controlpolicytostablysupportit. Foracontrolpolicy,stabilityregionisthesetof all traffic matrices that it can stably support. Network layer capacity region is defined as the union ofthestabilityregionsachievedbyallcontrolpolicies(possiblyunfeasible). Acontrolpolicy iscalledthroughput-optimalifitstabilizestheentirecapacityregion,meaningthatitsecures queuestabilityunderallstabilizabletrafficmatrices. 4.2.1 Systemdescription Consider a multiclass queuing network with set of nodesV, linksE and classesK that operates innormalizedtimeslotsn∈{0,1,2,···} . Foreachi∈V andd∈K,letq (d) i (n)betheinteger number ofd-packets in nodei at slotn. For each linkij∈E, letµ ij (n) be the link capacity at slotn,whichcountsthemaximumnumberofpacketsthelinkcantransmitatonetimeslotandis frequentlycalledlinktransmissionrateinliterature. Wealsodefinethelink actualtransmission f (d) ij (n) thatcountsthenumberofd-packetsgenuinely sentoverthelinkij atslotn. Itisimportanttodiscriminatebetweenlinkactualtransmissionandlinkcapacity. Inparticu- lar, while link capacities vary only by channel states, link actual transmissions are assigned by a network controllersubjectto 06f (d) ij (n)6min{q (d) i (n), µ ij (n)}. It is assumed that each packet leaves the network as soon as reaching its destination, and so thebacklogofd-packetsatthedestinationnodediszeroforalld∈K. Thestatevariablesof eachclassd canthenberepresentedbythevector q (d) ◦ (n):= q (d) 1 (n),...,q (d) d− 1 (n),q (d) d+1 (n),...,q (d) |V| (n) ∈R |V|− 1 88 MulticlassMinimumDelayRouting whereq (d) d (n)≡ 0 is dropped from the set of state variables. The set of state variables of the multiclasssystemisformedbyconcatenatingq (d) ◦ (n) ofdifferentclassesintoahypervectoras o q ◦ (n):=vec q (1) ◦ (n),...,q (|K|) ◦ (n) ∈R (|V|− 1)|K| . Notation 4.1. We use subscript◦ along with superscript(d) to denote reduced arrays obtained by discarding the entries corresponding to the destination noded. Subscript◦ on multiclass symbols,whichisnotaccompaniedbysuperscript(d)anymore,denotesreducedhyper-arrays obtainedbyconcatenatingallcorrespondingreducedarraysofdifferentclasses. Letastochasticprocessa (d) i (n)representtheintegernumberofexogenousd-packetsarriving intonodei atslotn. Thevectorofd-classarrivalsisobtainedas a (d) ◦ (n):= a (d) 1 (n),...,a (d) d− 1 (n),a (d) d+1 (n),...,a (d) |V| (n) ∈R |V|− 1 witha (d) d (n)≡ 0 beingdiscarded. The hypervectorofmulticlassnodearrivalsisformedas o a ◦ (n):=vec a (1) ◦ (n),...,a (|K|) ◦ (n) ∈R (|V|− 1)|K| We defined f (d) ij (n) as the integer number ofd-packets genuinely sent over the linkij at slot n. Letuscomposethevectoroflinkactualtransmissionsofd-packetsas f (d) (n):= f (d) 1 (n),...,f (d) |E| (n) ∈R |E| . Notethef (d) (n)isnotareducedvectorasnolinkisremovedfromnetworkbydiscardingthe destinationnoded. Thehypervectorofmulticlasslinkactualtransmissionsis f f(n):=vec f (1) (n),...,f (|K|) (n) ∈R |E||K| . RecallfromSec.3.4thatgivenadirectedgraph(V,E),wedenoteitsnode-edgeincidence matrix byB in which B iℓ is 1 if node i is the tail of directed edge ℓ, is− 1 if i is the head, andis0otherwise. Foraclassd,letB (d) ◦ denoteareductionofB throughdiscardingtherow correspondingtothedestinationnoded. WerefertoB (d) ◦ asthebasisincidencematrixwith respect to noded, or classd for that matter. Concatenating basis incidence matrices of different classes,themulticlassbasisincidencematrixisobtainedas IB ◦ :=diag B (1) ◦ ,...,B (|K|) ◦ ∈R (|V|− 1)|K|×|E||K| . (4.2) 4.2Preliminaries 89 OnecanthenverifythatIB ◦ f(n)isahypervectorinwhichtheentrycorrespondingtonodei andclassd isgivenby (IB ◦ f f) (d) i (n)= X b∈out(i) f (d) ib (n)− X a∈in(i) f (d) ai (n). Usingtheabovenotation,thef f-controlled,stochasticstatedynamicsofamulticlassqueuing network iscapturedby o q ◦ (n+1)= o q ◦ (n)+ o a ◦ (n)− IB ◦ f f(n). (4.3) Considering the difference between link capacity and link actual transmission explains why despitetraditionalnotationinliterature,wedonotneedany(·) + operationin (4.3). Under the assumption of no transmission to trapping nodes, if nodei is a trapping node for classd, then one can discardq (d) i (n) from the set of state variables o q(n),a (d) i (n) from the set of node arrivals o a(n),and therow correspondingtonodei fromthe basisincidence matrixB (d) ◦ . Note that in this case, nodei is not supposed to receive any exogenousd-packets. Further, it does notacceptd-packetsfromits incominglinks, andsoneithercanitsendanyd-packetson itsoutgoinglinks. 4.2.2 Networkcapacityregion In wirelesssystems,channelconditionsareuncontrollableparametersthatvaryintime dueto environmental change and user mobility. All information about topology state may be cast into an array whose entries at each timeslot describe channel state between every two corresponding nodes. We assume that the setsV andE change much slower than channel states so that we can fixthemduringthetimeofinterest. Nevertheless,thechannelconditionsmaylargelychange due to environmental conditions, wireless fading, power allocation and local mobility. We also assumethatchannelstatesremainfixedduringatimeslot,whiletheymaychangeacrossslots. Let a stochastic processS(n) = S 1 (n),··· ,S |E| (n) represent channel states at slot n, which describes all uncontrollable conditions that affect channel capacities. A temporarily unavailablechannel(dueto,e.g.,obstacleeffect)ismarkedwithzerocapacity. Supposethat S(n)evolvesaccordingtoanergodicstationaryprocessandtakesvaluesinafinite(butarbitrarily large) setS. Note that, for example, an irreducible Markov chain or any i.i.d. sequence of stochasticmatricesarebothergodicandstationary. ThenbyBirkhoff’sergodictheorem,each 90 MulticlassMinimumDelayRouting stateS∈S isofprobability s:=P S(n)=S =limsup τ →∞ 1/τ X τ − 1 n=0 I S(n)=S (4.4) with P S∈S s = 1. We insist that our proposed control policy does not require the state probabilitiess. However, the existence ofs is important to establish the network capacity region,andalsotocharacterizethestationaryrandomizedcontrolpolicies. Consider a connectivity graph(V,E) together with a channel state processS(n). For an arrival rate hypervector o a to be in the network capacity region, the necessary and sufficient condition is the existence of a sequence of joint link schedulingsπ (n) and link actual transmis- sionsf(n) such that their expected time averages mutually satisfy nodeflow conservation and linkcapacityconstraints,viz., a (d) i = X b∈out(i) f (d) ib − X a∈in(i) f (d) ai (4.5) X i∈V a (d) i = X a∈in(d) f (d) ad (4.6) X d∈K f (d) ij 6limsup τ →∞ 1/τ X τ − 1 n=0 π ij (n)E µ ij (n) (4.7) withtheoverbarnotationbeingdefinedin (4.1). Equalities (4.5)and (4.6)respectivelysecure flowconservation atintermediatenodes and at the destination d. Specifically, the equivalent matrix form of (4.5) is o a ◦ =IB ◦ f f, reading the expected time average of (4.3) subject to queue stability. Equality(4.6)guaranteesthatthereisnotrappingnode,andsoalld-classesarrivedinto thenetworkareultimatelycollectedinthedestinationnoded. Therighthandsideof (4.7)reads theeffectivecapacity ofeachlinkasπ ij µ ij . Thus,theinequalityguaranteesthelinkcapacity constraint subject to inter-channel interference that the average link hyperflow does not exceed theaveragepossiblecapacityavailableoneachlink. Note,inlightof (4.4),thattheexpected linkcapacitiesare E µ ij (n) = X S∈S sE µ ij (n) S(n)=S . The constraints (4.5)–(4.7) imply that the network capacity region is convex, closed and bounded[72]. Thus,if o a 1 isinthenetworkcapacityregion,sois o a 2 forany o a 2 4 o a 1 . Observethatthelinkactualtransmissionsarenotfixed,butdependonthecontrolpolicy. Also observe that there potentially exist infinite number of control policies that could meet theconstraints(4.5)–(4.7). Amongthemaretheonesthatusethesimpleprobabilityconcept of randomly distributing packets so that the desired time averages (4.5)–(4.7) are satisfied. 4.3MinimumDelayPolicy 91 As mentioned in Sec. 4.1, these stationary randomized policies typically require expensive computationalongwithperfectknowledgeofarrivalstatisticsandchannelstateprobabilities that are prohibitive in practice. Nevertheless, the existence of these queue-independent policies playsacrucialroleinouranalysis. 4.2.3 MulticlassBack-Pressure(BP)policy Consider a multiclass network where each node may have packets for different destinations. At every timeslot n, original multiclass BP [44] observes queue backlogs q (d) i (n) and estimates channel capacitiesµ ij (n) tomakeatransmissiondecisionasfollows. 1) Multiclass BP weighting: On every directed linkij and for eachclassd find the link queue- differential q (d) ij (n):=q (d) i (n)− q (d) j (n) andselecttheoptimalclass d ∗ ij (n):=argmax d∈K q (d) ij (n). (4.8) Thengiveaweighttothelinkusingitsestimatedcapacityas w ij (n):=µ ij (n)q (d ∗ ) ij (n) + . (4.9) 2) MulticlassBPscheduling: Findtheschedulingvectorsuchthat π (n)=argmax π ∈Π X ij∈E π ij w ij (n) (4.10) wheretiesarebrokenrandomly. 3) MulticlassBPforwarding: Overeachactivatedlinkij withw ij (n)>0transmitpacketsfrom the classd ∗ ij (n) at full capacityµ ij (n). If there is no enough packets from the classd ∗ ij (n) at nodei,transmitnullpackets. 4.3 MinimumDelayPolicy ToprepareaconvenientwayofunifyingourproposedcontrolwiththepreviousworksonBP schemes,ourproposedcontrolpolicyisdesignedwiththesamealgorithmicstructure,complexity andoverheadasBP.ThisprovidesaneasywaytoleveragealladvancedimprovementstoBP (usinge.g. LIFOservice,packetages,adaptiveredundancy,queueprioritization,etc.) tofurther 92 MulticlassMinimumDelayRouting enhancethenewcontrolpolicy. Italsosimplifiestheapproachtopracticeviaasmoothsoftware transitionfromBPtothenewpolicy. Weintroducethemulticlasscontrolpolicyintwoversions: (SCLT) Singleclassperlinkpertimeslot,whereeachactivatedlinktransmitsfromonlyone classateachtimeslot—whichisthewaythatBPperforms. (MCLT) Multipleclassesperlinkpertimeslot,whereeachactivatedlinkoptimallytransmits fromdifferentclassesateachtimeslot. 4.3.1 MulticlassminimumdelaysubjecttoSCLT Consider a multiclass network subject to SCLT assumption, where each link may transmit packetsfromonlyoneclassateachtimeslot—thesamewaythatmulticlassBPperforms. Under SCLTassumption,theminimumdelaypolicyobservesqueuebacklogsq (d) i (n)andestimates channelcapacitiesµ ij (n) tomakeatransmissiondecisionasfollows. 1) Multiclass minimum delay weighting under SCLT: On every directed linkij and for each class d find the link queue-differential q (d) ij (n) := q (d) i (n)− q (d) j (n) and select the optimal class, in the same way as BP, using (4.8). Next, calculate the the link actual transmission prediction,whichisthenumberofpacketsthelinkwouldtransmitifitwereactivated,as ̂ f (d ∗ ) ij (n):=min q (d ∗ ) ij (n) + , µ ij (n) . (4.11) wherethehatnotationdenotesapredictedvaluewhichwouldnotnecessarilyberealized. Thengiveaweighttothelinkusingitspredictedactualtransmissionas w ij (n):=2q (d ∗ ) ij (n) ̂ f (d ∗ ) ij (n)− ̂ f (d ∗ ) ij (n) 2 . (4.12) 2) MulticlassminimumdelayschedulingunderSCLT:Findtheschedulingvector,inthesame wayas BP,usingthemax-weight scheduling(4.10). 3) Multiclass minimum delay forwarding under SCLT: Transmit ̂ f (d ∗ ) ij (n) number of packets fromtheclassd ∗ ij (n) overeachactivatedlinkij,leadingto f (d) ij (n)=    ̂ f (d ∗ ) ij (n) if π ij (n)=1 and d=d ∗ ij (n) 0 otherwise. (4.13) 4.3MinimumDelayPolicy 93 Itiscriticaltodiscriminateamonglinkactualtransmissionf (d) ij (n),linktransmissionpre- diction ̂ f (d) ij (n) and link capacityµ ij (n). Also notice that ̂ f (d) ij (n) in (4.13) could be non-integer. In practice, the number of packets to be transmitted over links can be rounded to the nearest integer withno important influenceon the performance. To be more precise, however, every nodemayalgebraicallyaddthepacketresidualssentoneachofitsongoinglinksfordifferent classessothatmakingacompensationassoonasthesumhitseither1 or− 1. Remark4.1. Ifq (d ∗ ) ij (n)60,weget ̂ f (d ∗ ) ij (n)=0,whichinturnleadstow ij (n)=0,meaningthat evenifthelinkij werescheduled,stillnopacketwouldbetransmittedoverit. Ifq (d ∗ ) ij (n)>0,on the other hand, we getq (d ∗ ) ij (n) + =q (d ∗ ) ij (n), which together with ̂ f (d ∗ ) ij (n)6q (d ∗ ) ij (n) due to(4.11), meansthatthelinkweight (4.12)isalwaysnon-negative. Further,thefactq (d ∗ ) ij (n) + 6q (d ∗ ) i (n) impliesthat ̂ f (d ∗ ) ij (n) neverexceedsthenumberofd ∗ -packetsinthetransmittingnode. Remark4.2. WhileBPisdrivenbylinkcapacitiesµ ij (n),theminimumdelaypolicyemphasizes actualnumberoftransmittablepacketsf (d) ij (n)subjecttothelinkcapacity. Thus,weallocate available resources based only on genuinely transmittable packets, without counting on null packetsasbeingpracticedinBPschemes. Remark 4.3. The minimum delay policy takes the link weight w ij (n), which itself directly controlsthemax-weightscheduling,quadraticinlinkqueue-differential,wherefor q (d ∗ ) ij (n)6 µ ij (n)issimplifiedto w ij (n)= q (d ∗ ) ij (n) 2 . ThiscontrastswithBPweightingwhichislinearin link queue-differential. The quadraticweight is a keycharacteristic that leadsto the minimum routingdelayproperty,discussedinSec.4.6. Remark 4.4. UnlikeBPthatforwardsthemaximumpossiblenumberofd ∗ -packetsovereach activated link, the minimum delay policy limits packet forwarding to link queue-differential. Besidesformingaquadraticlinkweightinlinkqueue-differential,thisreduces queueoscillations bythedecreaseofunnecessarypacketforwardingsacrossthelinks. Remark 4.5. Like BP, also the minimum delay policy is based on a centralized scheduling whosecomplexityisprohibitiveinpractice. However,muchprogresshasrecentlybeenmade toeasethisdifficultybyderivingdecentralizedschedulerswiththeperformanceofarbitrarily closetothecentralizedversion[86,61,62]. Remark 4.6. In packet switches, the work of [45] extends queue-based scheduling to admit more general functions of queue lengths with a particular interest onα -weighted schedulers that useα -exponent of queue lengths. There has been a non-proven conjecture that in heavy 94 MulticlassMinimumDelayRouting traffic condition, average delay is minimized when α →0. A discussion of this was given in [58]alongwithsomecounterexamples. Inaveryspecialcasewhenalllinkcapacitiesarethe same, i.e.,µ ij (n)=µ (n),∀ij∈E,andalllinkqueue-differentialsarealwayslessthanit,i.e., q (d ∗ ) ij (n)<µ (n),∀ij∈E,ourproposedminimumdelaypolicyunderSCLTassumptionturns to be equivalent toα -weighted policy withα =2. However, the requirement ofq ij (n)<µ ij (n) ina multihoprouting problemwouldimply thatthe networkisnot inaheavy trafficcondition. Thus,eveniftheconjectureweretrue,itdoescausenocontradictionwithourresultshere. 4.3.2 MulticlassminimumdelaywithMCLT It seems obvious that network resources are squandered by restricting the control policy to transmitonlyoneclassofpacketsperlinkpertimeslot. Inotherwords,thelargercapacityof networkwouldbeutilizedateachtimeslot,andsotheaveragenetworkdelaywoulddecrease, if each activated link were properly filled up to its full capacity. At the same time, we will analyticallyshowinSec.4.6thatblindlyfillingupthelinksbysimplysendingthemaximum number of packets from only one selected class, which is basically practiced in BP schemes, only depletes the network resources with even negative impact on delay performance. Thus, theimportantquestionis,howadynamicroutingcontrolwithnoroutingpathconstraintcan minimizetheaveragenetworkdelaybymaximizingresourceutilizationateachtimeslot. We answer the above question by revising the minimum delay algorithm to optimally de- termine the number of packets from different classes, rather than from only one class, that each activated link should transmit at each timeslot. Note that the algorithmic structure and implementationcomplexityremainsthesame. Dropping SCLT assumption, the minimum delay policy observes queue backlogsq (d) i (n) andestimateschannelcapacitiesµ ij (n) tomakeatransmissiondecisionasfollows. 1) Multiclass minimum delay weighting with MCLT: On every directed linkij and for each classd find the link queue-differential q (d) ij (n):=q (d) i (n)− q (d) j (n) and determine ̂ f (d) ij (n) by solvingthefollowinglocalconstrainedoptimizationproblem: ̂ f (d) ij (n):=argmin X d∈K q (d) ij (n)− ̂ f (d) ij (n) 2 s.t. 1) P d∈K ̂ f (d) ij (n)6µ ij (n) 2) 06 ̂ f (d) ij (n)6q (d) i (n), ∀d∈K (4.14) 4.3MinimumDelayPolicy 95 ( , ) (1) ij f q (2) 5  (1) ij f Unique optimal solution in the absence of variable bounds and integer constraints Two optimal solutions in the presence of variable bounds and integer constraints () n (2) ij f () n () n (2) ij f  ij   () n () n ij () n q (2) ij () n q (1) ij () n 6  ij  () n q (1) 4  ij () n 6  ij  () n Fig.4.1 Geometricaldescriptionof problem (4.14) for a two-class case withq (1) ij (n)+q (2) ij (n)> µ ij (n), showing that the integer problemcanhavemorethanoneoptimalsolution. Thengiveaweighttoeachclassd as w (d) ij (n):=2q (d) ij (n) ̂ f (d) ij (n)− ̂ f (d) ij (n) 2 (4.15) andaggregatethemtodeterminethefinallinkweightas w ij (n):= X d∈K w (d) ij (n). (4.16) 2) Multiclassminimumdelayscheduling with MCLT:Find theschedulingvector,in thesame wayasBP,usingthemax-weightscheduling(4.10). 3) MulticlassminimumdelayforwardingwithMCLT:Transmit ̂ f (d) ij (n)numberofpacketsfrom theclassd overeachactivatedlinkij. Table4.1comparesmulticlassBPalgorithmwiththeproposedminimumdelayalgorithmin two scenarios of being subject to either transmit only one class orallowed to transmit multiple classes per link per timeslot. The emphasis is on the same structure, computational complexity andoverheadsignalingofallthreealgorithms. Remark 4.7. On uniclass networks, both minimum delay policies under SCLT and MCLT becomethesameasParetooptimalHDpolicywithβ =0,proposedinCh.3. 96 MulticlassMinimumDelayRouting Table4.1Comparing algorithmic structureofmulticlassBP,multiclassminimumdelayunder SCLT (MD-SCLT) and multiclassminimumdelayunderMCLT(MD-MCLT)policies. Weighting Multiclass BP d ∗ ij (n)=argmax d∈K q (d) ij (n) ̂ f (d) ij (n)= ( min{µ ij (n), q (d) i (n)} if d=d ∗ ij (n) 0 otherwise w ij (n)=µ ij (n)q (d ∗ ) ij (n) + MD-SCLT d ∗ ij (n)=argmax d∈K q (d) ij (n) ̂ f (d) ij (n)= ( min{q (d ∗ ) ij (n) + , µ ij (n)} if d=d ∗ ij (n) 0 otherwise w ij (n)=2q (d ∗ ) ij (n) ̂ f (d ∗ ) ij (n)− ̂ f (d ∗ ) ij (n) 2 MD-MCLT ̂ f (d) ij (n)=argmin X d∈K q (d) ij (n)− ̂ f (d) ij (n) 2 s.t. P d ̂ f (d) ij (n)6µ ij (n) & 06 ̂ f (d) ij (n)6q (d) i (n) w ij (n)= X d∈K 2q (d) ij (n) ̂ f (d) ij (n)− ̂ f (d) ij (n) 2 Scheduling π (n)=argmax π ∈Π X ij∈E π ij w ij (n) Forwarding f (d) ij (n)=    ̂ f (d) ij (n) if π ij (n)=1 0 otherwise Optimizationproblem (4.14)isastandardleast-normproblemwithvariableboundsthatcan be solved in fast polynomial time at each node, i.e., in a fully decentralized manner throughout thenetwork. Asuggestedsolutionisprovidedasfollows. To simplify the notation, we drop the time variable(n). First observe that in solving (4.14), it is trivial that one needs to take ̂ f (d) ij =0 for each classd withq (d) ij 60. Thus, let us create the setK ij ⊆K such thatq (d) ij >0,∀d∈K ij and fix ̂ f (d) ij =0,∀d / ∈K ij . Then observe that in solving(4.14),thefollowingisalsotrue: If X d∈K ij q (d) ij 6µ ij thenoptimal ̂ f (d) ij =q (d) ij , ∀d∈K ij . Thus,weonlyneedtosolvetheproblemfor P d∈K ij q (d) ij >µ ij ,whichconvertsthefirstconstraint in (4.14) from inequality to equality. Then in the absence of variable bounds and integer 4.3MinimumDelayPolicy 97 constraints,usingabasicLagrangeargument,theproblemhasauniquesolutionas ̂ f (d) ij =q (d) ij + µ ij − X d∈K ij q (d) ij |K ij |, ∀d∈K ij . (4.17) Fromageometricalstandpoint,(4.17)representstheprojectionofpoint(q (1) ij ,··· ,q (|K ij |) ij ) ontothehyperplane P d∈K ij ̂ f (d) ij =µ ij . Underintegerconstraints,thishyperplaneistransformed into an integer hypergrid, where the optimal solution(s) will be the vertex(es) of this hypergrid withthe shortest Euclideandistance to the point(q (1) ij ,··· ,q (|K ij |) ij ). Note that the solutionto the integer problem is not necessarily unique. Adding the variable bounds into the picture, thesolutionhastobealsointeriorto,oronto,theintegerbox06f (d) ij 6q (d) ij ,∀d∈K ij . The geometricalprocedureisdescribedbyFig.4.1foratwo-classcase. Thecorrespondingalgorithm forsolvingproblem (4.14)issummarizedinAlg.4.1. Algorithm4.1Solvingproblem (4.14)foralinkij atslotn. CreateasetK ij (n)⊆K suchthatq (d) ij (n)>0,∀d∈K ij (n) Assign ̂ f (d) ij (n)=0,∀d / ∈K ij (n) if P d∈K ij (n) q (d) ij (n)6µ ij (n) S1: Assign ̂ f (d) ij (n)=q (d) ij (n), ∀d∈K ij (n) else S2: Leth:= P d∈K ij (n) q (d) ij (n)− µ ij (n) |K ij (n)| andset ̂ f (d) ij (n)=q (d) ij (n)− h, ∀d∈K ij (n) S3: Foranyd ′ ∈K ij (n) with ̂ f (d ′ ) ij (n)<0,assign ̂ f (d ′ ) ij (n)=0, removed ′ fromK ij (n),andgobacktoS2 S4: Letr :=µ ij (n)− P d∈K ij (n) ̂ f (d) ij (n) andassign ̂ f (d) ij (n)=    ̂ f (d) ij (n) for r randomlychosenclassesin K ij (n) ̂ f (d) ij (n) fortherestof classesin K ij (n) end Observe that S2 finds the optimal solution in the absence of variable bounds and integer constraints, S3 ensures that the solution meets the variable bounds, and S4 determines an integer solution by finding a vertex on the integer hypergrid—the first constraint of (4.14) with equality—with the shortest distance to the the initial solution obtained by S2–S3. In case of 98 MulticlassMinimumDelayRouting discardingd ′ fromK ij (n) inS3,westillget X d∈K ′ ij (n) q (d) ij (n)>µ ij (n) with K ′ ij (n):=K ij (n)−{ d ′ }. Toconfirmthis,observethat ̂ f (d ′ ) ij (n)<0 implies |K ′ ij (n)|q (d ′ ) ij (n)< X d∈K ′ ij (n) q (d) ij (n)− µ ij (n) whichleads to P d∈K ′ ij (n) q (d) ij (n)>µ ij (n) becauseq (d ′ ) ij (n)>0,and sothe lefthandside ofthe latterinequalityispositive. The term “r randomly chosen classes” in S4 appears due to the fact that the integer problem mayhavemorethanonesolution. WhentheinitialsolutionofS2–S3isinteger,itwillbethe solutiontotheintegerproblemtooandsoisunique. Otherwise,therepotentiallyexistseveral verticesontheintegerhypergridwithequaldistancesfromtheinitialuniquenon-integersolution andshorterthanthedistancesofallothervertices(seeFig.4.1). 4.4 DynamicCharacteristic Givennetworkconditionateachtimeslot,dynamiccharacteristicofacontrolpolicyidentifies its action on packet forwarding, which is a key to analyze traffic behavior. We formalize such a characteristicforourminimumdelaypolicyinthenexttheorem,whichiscoretotheproofof Th.4.3onroutingdelayminimizationandTh.4.2onnetworkthroughputoptimality. Theorem 4.1. (Dynamic Characteristic) At every timeslot n, the minimum delay policy maximizesthef f-controlledfunctional G(f f, o q ◦ ,n):=2f f(n) ⊤ IB ◦ ⊤ o q ◦ (n)− f f(n) ⊤ IB ◦ ⊤ IB ◦ f f(n) (4.18) subject to network constraints, including directionality, capacity and interference. Specifically, thefollowingstatementsaretrue: (SCLT) On a multiclass network subject to transmitting from at most one class per link per timeslot,G(f f, o q ◦ ,n) ismaximizedbytheminimumdelaypolicyofSec.4.3.1. (MCLT) On a multiclass network without restricting the transmission to only one class per linkpertimeslot,G(f f, o q ◦ ,n) ismaximizedbytheminimumdelaypolicyofSec.4.3.2. 4.4DynamicCharacteristic 99 Proof. Weoftendroptimeslotvariable(n)foreaseofnotation. Letustemporarilyignoreall network constraints including interference, capacity, directionality and integer restriction on f f(n). We later integrate the effect of these constraints into the solution. With no constraint, f f(n) is admissible iff P b∈out(i) f (d) ib (n)6q (d) i (n). The vector spaceF formed by the collection ofalladmissiblelinkactualtransmissionsf f(n)isaconvexset. Toseewhy,observethatforany twoadmissiblef f 1 (n),f f 2 (n)∈F andforanyscalarc∈[0,1],theflow cf f 1 (n)+(1− c)f f 2 (n) isalsoadmissibleandsoisinF. Next,weshowthatD(f f) isconcaveonF,orequally G cf f 1 +(1− c)f f 2 >cG(f f 1 )+(1− c)G(f f 2 ). ExpandingG,followedbysomematrixalgebra,weneedtoshow c(1− c) IB ◦ f f 1 − IB ◦ f f 2 ⊤ IB ◦ f f 1 − IB ◦ f f 2 >0 which is trivial. Having shown that G(f f) is concave, solution to the G(f f) maximization problemisobtainedfrom∇ f f D(f f)=0,leadingtoIB ◦ ⊤ IB ◦ f f =IB ◦ ⊤ o q ◦ . Letusdefine γ (d) ij (n):= IB ◦ ⊤ IB ◦ f f(n) (d) ij which represents the entry of hypervectorIB ◦ ⊤ IB ◦ f f corresponding to the edgeij and classd at slotn. Considering(IB ◦ ⊤ o q ◦ ) (d) ij =q (d) ij , theG maximizing conditionIB ◦ ⊤ IB ◦ f f =IB ◦ ⊤ o q ◦ entails γ (d) ij (n)=q (d) ij (n) foreachlinkij andeachclassd. Assumethatacontrolpolicytakesα ij f (d) ij =q (d) ij forscalarsα (d) ij >0,whichwewilldeter- mine them later. By the argument above, such a control policy can maximize G(f f) at each timeslot,ifitcomplieswithγ (d) ij =α (d) ij f (d) ij . Pluggingthis in (4.18),maximumG(f f) isfound as G max = X ij∈E X d∈K f (d) ij 2q (d) ij − α (d) ij f (d) ij . (4.19) Inlightoftheforwardinglawα (d) ij f (d) ij =q (d) ij ,thelatterleadsto G max = X ij∈E X d∈K q (d) ij 2 α (d) ij (4.20) whichrepresentsthemaximumofG(f f)forspecificvaluesof α (d) ij >0andintheabsenceofall network constraintsincludingdirectionalityandcapacityofairlinks. Let us add network constraints into the solution. Obviously, theG max as (4.20) is no longer attainable. However, going one step back to (4.19), under the forwarding lawα (d) ij f (d) ij =q (d) ij , 100 MulticlassMinimumDelayRouting maximizingf f at each slotn can be found from the following integer optimization problem, wherethefirstconstrainttreatslinkdirectionality,thesecondconstraintensureslinkcapacity, thethirdconstraintreadsthefactthatthenumberofd-packetsleavinganodecannotbelarger thanthequeuebacklogofclassdatthatnode,andthefourthconstraintimposestherequirement ofinterferencemodelonsimultaneousactivationofdifferentlinks: max f (d) ij (n) X ij∈E X d∈K 2q (d) ij (n)f (d) ij (n)− α (d) ij f (d) ij (n) 2 s.t. 1) f (d) ij (n)>0, ∀d∈K 2) P d∈K f (d) ij (n)6µ ij (n) 3) P b∈out(i) f (d) ib (n)6q (d) i (n), ∀d∈K 4) Interference Model. (4.21) Defining o α as the hypervector of coefficients α (d) ij , let us first determine such o α that maxi- mizes(4.21). Obviously,themaximizingα (d) ij isthesmallestone. However, o α islowerbounded bythefollowingsystemrestrictions: f f ⊤ f f (a) 6 IB + ◦ f f ⊤ IB + ◦ f f (b) 6 o q ◦ ⊤ o q ◦ (c) = f f ⊤ IB ◦ ⊤ IB ◦ f f (d) = f f ⊤ diag(o α )f f whichleadstothemaximizingdiag(o α )=I. Theinequality(a) readsthefactthat X i∈V X d∈K X b∈out(i) f (d) ib 2 6 X i∈V X d∈K X b∈out(i) f (d) ib 2 . NotethatIB + ◦ representstheincidencematrixwithall− 1entriesbeingreplacedby0. Thus, IB + ◦ f f isahypervectorinwhichtheentrycorrespondingtonodeiandclassdshowstheoutgoing flowofclass dfromthenode. Thentheinequality(b)readsthefactthatateachtimeslot,the numberofd-packetsleavinganodeiisatmostequaltothequeuebacklogofclassdatthenode. Theidentity(c)resultsfromtheGmaximizingconditionIB ◦ ⊤ IB ◦ f f =IB ◦ ⊤ o q ◦ . Toconfirmthis, multiplybothsidesbyIB ◦ toget(IB ◦ IB ◦ ⊤ )IB ◦ f f =(IB ◦ IB ◦ ⊤ ) o q ◦ ,wherethepositivedefiniteness ofIB ◦ IB ◦ ⊤ provides the result. The identity(d) readsIB ◦ ⊤ IB ◦ f f =diag(o α )f f that follows from the forwarding lawα (d) ij f (d) ij =q (d) ij , which has the matrix equivalence ofdiag(o α )f f = IB ◦ ⊤ o q ◦ subjecttotheG maximizingconditionIB ◦ ⊤ IB ◦ f f =IB ◦ ⊤ o q ◦ . Notethatsincethewirelessnetworkhasaconnectedtopology,anysub-matrixobtainedfrom itsincidencematrixbyremovinganyarbitraryrowisoffullrowrank(seeSec.2.3). Thus,B (d) ◦ 4.4DynamicCharacteristic 101 hasfullrowrankforalld∈K. Thus,thehypermatrixIB ◦ isablockdiagonalmatrixoffullrow rank matricesand so is full row rank. This implies that the symmetric positive semi-definite matrixIB ◦ IB ◦ ⊤ hasfullrankandsoispositivedefinite. Now, we show that under either SCLT or MCLT assumptions, the proposed minimum delay policies of either Sec. 4.3.1 or 4.3.2 respectively solves the optimization problem (4.21) for α (d) ij = 1. Assuminganinterferencemodelunderwhichanodecannotconcurrentlytransmit to more than one neighboring node at one timeslot, the third constraint in (4.21) reduces to f (d) ij (n)6q (d) i (n),∀d∈K. Thisway,onecandecoupletheimpactofinterferencefromother constraints. In other words, it allows to solve the centralized optimization problem (4.21) in two independent steps of first locally weighting each link and then globally scheduling a set of links withmaximumtotalweight. Lookinginto(4.21)forα (d) ij =1,thelocallinkweighting is formalizedbythedistributed optimizationproblem w opt ij (n):=max f (d) ij (n) X d∈K 2q (d) ij (n)f (d) ij (n)− f (d) ij (n) 2 s.t. 1) P d∈K f (d) ij (n)6µ ij (n) 2) 06f (d) ij (n)6q (d) i (n), ∀d∈K (4.22) thatfindstransmissionpredictionsforeachoutgoinglink ij connectedtothenodei. Thenthe globalmax-weightschedulingisformalizedbythecentralized optimizationproblem π opt (n):=argmax π ∈Π X ij∈E π ij w opt ij (n) (4.23) that finds the set of activated links with maximum total weight. Observing that (4.23) reads the max-weight scheduling, it remains to show thatw ij (n) in (4.11)–(4.12) under SCLT, and in (4.14)–(4.16)withMCLT,complywithw opt ij (n) in (4.22). Under SCLT, where each link must choose to transmit from only one class at each timeslot, itistrivialthattheoptimalclasstosolve(4.22)istheonewiththelargestlinkqueue-differential, givenby (4.8),whichsimplifies(4.22)to w opt ij (n)=max f (d ∗ ) ij (n) 2q (d ∗ ) ij (n)f (d ∗ ) ij (n)− f (d ∗ ) ij (n) 2 s.t. 1) f (d ∗ ) ij (n)6µ ij (n) 2) 06f (d ∗ ) ij (n)6q (d ∗ ) i (n) (4.24) 102 MulticlassMinimumDelayRouting For each linkij, first derivative with respect to f (d ∗ ) ij leads to the maximizing link transmission f (d ∗ ) ij =q (d ∗ ) ij 6q (d ∗ ) i . Applyingtheconstraints,weget w opt ij (n)=2q (d ∗ ) ij (n)f (d ∗ ) ij (n)− f (d ∗ ) ij (n) 2 f (d ∗ ) ij (n)=min q (d ∗ ) ij (n) + , µ ij (n) whichfollows(4.11)–(4.12)underSCLTassumption. With MCLT, where a link could transmit from different classes at one timeslot, first observe thatgivencurrentq (d) ij (n),thefollowingidentityistrue: arg min f (d) ij (n) X d∈K q (d) ij (n)− f (d) ij (n) 2 =arg max f (d) ij (n) X d∈K 2q (d) ij (n)f (d) ij (n)− f (d) ij (n) 2 Then it is easy to confirm that w ij (n) derived by (4.14)–(4.16) complies withw opt ij (n) assigned bytheoptimizationproblem (4.22). Remark 4.8. Onuniclassnetworks,G(f f, o q ◦ ,n)reducestotheG(f,q ◦ ,n)functional (3.31) inCh.3,whichismaximizedbythe ParetooptimalHDpolicywithβ =0. 4.5 ThroughputAnalysis Toanalyzethestabilityofourminimumdelaypolicy,weexploitthetheoryofLyapunovdrift forstochasticdiscrete-timesystems. ConsidertheclassicalquadraticLyapunovcandidate W(n):= o q ◦ (n) ⊤ o q ◦ (n)= X ij∈E X d∈K q (d) i (n) 2 . LettingtheLyapunovdrift∆ W(n):=W(n+1)− W(n) andsubstitutingforthestatehyper- vector o q ◦ (n+1) from (4.3)leadto ∆ W(n)=2 o a ◦ (n)− IB ◦ f f(n) ⊤ o q ◦ (n)+ o a ◦ (n) ⊤ o a ◦ (n) +f f(n) ⊤ IB ◦ ⊤ IB ◦ f f(n)− 2f f(n) ⊤ IB ◦ ⊤ o a ◦ (n). (4.25) UsingtheG(f f, o q ◦ ,n) expressionof (4.18)yields ∆ W(n)= 2 o a ◦ (n) ⊤ o q ◦ (n)− G(f f, o q ◦ ,n)+ o a ◦ (n) ⊤ o a ◦ (n)− 2f f(n) ⊤ IB ◦ ⊤ o a ◦ (n). (4.26) 4.5ThroughputAnalysis 103 Now consider a multiclass arrival rate o a ◦ being interior to the capacity regionC, i.e., there exists a vectorϵ with positive entries such that o a ◦ +ϵ ∈ C. Thus, by condition (4.5), there exists such a hyperflow f f ′ (n) as IB ◦ f f ′ = o a ◦ +ϵ . At the same time, Th. 4.1 guarantees that G(f f ⋆ , o q ◦ ,n)> G(f f ′ , o q ◦ ,n) at each slot n, where f f ⋆ (n) represents the multiclass link actual transmissions provided by the minimum delay policy at slot n under either SCLT or MCLT assumptions. Then the next theorem is proven by showing that the expected value of the Lyapunovdrift (4.25)isalwaysnegative. Theorem4.2. (ThroughputOptimality)Onmulticlasswirelessnetworks,theminimumdelay policy is throughput-optimal in the sense that it guarantees network stability for any traffic matrixinthenetworkcapacityregion,definedbytheconstraints(4.5)–(4.7). Proof. Tosimplifytherestoftheproof,weassumearrivalsarei.i.d. overtimeslots,withthe understandingthatitcaneasilybemodifiedto yieldsimilarresultfornon-i.i.d. arrivals, using theN-slotanalysisderivedfrom (3.29). We often drop timeslot variable(n) for ease of notation. Taking conditional expectation from (4.26)giventhecurrentqueuebacklogs o q ◦ (n) yields E ∆ W|o q ◦ =2E o a ◦ ⊤ o q ◦ o q ◦ − E G(f f) o q ◦ + E o a ◦ ⊤ o a ◦ − 2f f ⊤ IB ◦ ⊤ o a ◦ o q ◦ (4.27) where the expectation is with respect to the randomness of arrivals, channel states, and routing decision—incaseofarandomizedroutingalgorithm. Letf f ⋆ (n)bethemulticlasslinkactualtransmissionsprovidedbyourproposedcontrolat slotn,andf f(n)betheonesprovidedbyanyalternativecontrolleratthesametimeslot. Inlight ofTh.4.1,foreachslotn wehaveG(f f ⋆ , o q ◦ ,n)>G(f f, o q ◦ ,n),viz., G(f f ⋆ , o q ◦ ,n)>2f f(n) ⊤ IB ◦ ⊤ o q ◦ (n)− f f(n) ⊤ IB ◦ ⊤ IB ◦ f f(n). Takingconditionalexpectationleadsto E G(f f ⋆ ) o q ◦ >2E f f ⊤ IB ◦ ⊤ o q ◦ o q ◦ − E f f ⊤ IB ◦ ⊤ IB ◦ f f o q ◦ foranyalternativetransmissiondecisionf f(n). Thisincludesthecasewheref f(n)isproducedby a routing algorithm that makes independent, stationary and randomized transmission decisions ateachslotnbasedonlyoncurrentlinkcapacitiesandsoindependentofqueuelengths[72]. Let us fix f f(n) for such an algorithm and call it f f ′ (n). By independency of f f ′ (n) from 104 MulticlassMinimumDelayRouting o q ◦ , we getE{f f ′⊤ IB ◦ ⊤ |o q ◦ } = E{f f ′⊤ IB ◦ ⊤ }. Since o a ◦ is interior to the capacity regionC, one can determine such f f ′ (n) that stabilizes an arrival rate o a ◦ +ϵ ∈ C at each slot n. By the feasibilitycondition (4.5)andthei.i.d. assumptiononarrivalsandchannelstates,wethenget E{IB ◦ f f ′ }= o a ◦ +ϵ , implying that both o a ◦ andf f ′ reach their steady states on each and every timeslot. Thus,E{f f ′⊤ IB ◦ ⊤ |o q ◦ }=(o a ◦ +ϵ ) ⊤ . UsingthisintheconditionalexpectationofG(f f ⋆ ) yields E G(f f ⋆ ) o q ◦ >2(o a ◦ +ϵ ) ⊤ o q ◦ − E f f ′⊤ IB ◦ ⊤ IB ◦ f f ′ o q ◦ . (4.28) Sincethenetworklayercontrollerhasnoimpactonarrivals, o a ◦ (n)isanindependentsystem variable,whichimpliesE{o a ◦ ⊤ |o q ◦ }=E{o a ◦ ⊤ }= o a ◦ withthelastequalitybeingduetothei.i.d. arrivals. Exploitingthisand (4.28)inthedriftinequality (4.27)yields E ∆ W|o q ◦ 6− 2ϵ ⊤ o q ◦ +E Γ |o q ◦ (4.29) Γ:= o a ◦ ⊤ o a ◦ − 2f f ⋆⊤ IB ◦ ⊤ o a ◦ +f f ′⊤ IB ◦ ⊤ IB ◦ f f ′ − f f ⋆⊤ IB ◦ ⊤ IB ◦ f f ⋆ . InvestigatingΓ( n), note that all arrivals have finite mean and variance, and that each link actual transmissionisatmostequaltothelinkcapacitywhichisfinite,andsoboth f f ⋆ (n)andf f ′ (n) have finite upper bounds. Thus, the expected value of Γ( n) is finite at each slot n, and so there existssuchaΓ max >0 asE{Γ |o q ◦ }6Γ max . Usingthisin (4.29)yields E ∆ W|o q ◦ 6− 2ϵ ⊤ o q ◦ +Γ max . Define ϵ min as the smallest entry of the ϵ vector. Then for ∥o q ◦ ∥ > Γ max (2ϵ min ) we get E{∆ W|o q ◦ }<0. Thus, by Theorem 2 in [51], the queuing system is stable, and so o a ◦ belongs tothestabilityregionofourminimumdelaypolicy. Sinceϵ canbearbitrarilysmall,thisimplies thatanytrafficmatrixbeinginteriortothenetworkcapacityregionisstabilizedbytheminimum delaypolicy,whichprovesitsthroughputoptimality. Remark 4.9. AcommonthemetoalloftheworksonBP,goingbacktotheoriginalpaper[44], isthatthealgorithmisderivedbythegreedyminimizationofaboundontheLyapunovdrift. As aresult,theBPschedulingisformulatedbasedonlinkcapacitiesandtheBPforwardingspreads the maximum number of packets along the activated links. To the best of our knowledge, this is thefirsttimeanetworkcontrollergenuinelyminimizestheLyapunovdrift—viamaximizing theG(f f, o q ◦ ,n) functional—rather thanmerely pushingdownan upper-bound on thedrift. As a result, our proposed scheduling is formulated based on link actual transmissions and our proposedforwardingiscontrolledbylinkqueue-differentials. 4.6RoutingDelayAnalysis 105 4.6 RoutingDelayAnalysis As the main result, this section shows that our proposed routing control minimizes average network delay in the class of all control algorithms that act based only on current queue conges- tion and current channel states. Precisely, it is shown that in this class of control algorithms, the minimum delay policy solves the following optimization problem without requiring the knowledgeofnetworktopology,arrivalstatisticsandchannelstateprobabilities: min Q:= X i∈V X d∈K q (d) i s.t. Network Constraints. (4.30) Remark 4.10. By Little’s Theorem [83], for a given traffic matrix, expected time average total queue congestion Q is proportional to long-term average end-to-end network delay. Hence, minimizingQ indeedensuresminimizingaveragenetworkdelay. NextassumptionisusedintheproofofTh.4.3,statingthatthetimeslotmaximizationof G(f f, o q ◦ ,n) is an optimal substructure for its long-term average maximization, which means no overlapping among substructures. Thus, to solve theG maximization problem, one can greedily maximizeG at each slotn. Worth to note that in practice, this is a mild assumption that can be metby mostwirelessmeshnetworks. Assumption4.1. Givenacombinationofnetworktopology,trafficratesandnetworkconstraints, timeslotmaximizationofG(f f, o q ◦ ,n)providesanoptimalsubstructureforglobalmaximization ofG(f f, o q ◦ ):=2f f ⊤ IB ◦ ⊤ o q ◦ − f f ⊤ IB ◦ ⊤ IB ◦ f f . Theorem4.3. (MulticlassMinimumDelay)LetamulticlasswirelessnetworkmeetAssum.4.1 underastabilizabletrafficmatrix. Consideraclassofnetworkcontrollersthatactbasedonlyon current queue congestion and current channel states, including the ones with perfect probability knowledgeofarrivalsandchannelstates. Withinthisclass,theminimumdelaypolicysolves the network delay minimization problem (4.30). Specifically, the following statements are true: (SCLT) On a multiclass network subject to transmitting from at most one class per link per timeslot,Q isminimizedbytheminimumdelaypolicyofSec.4.3.1. (MCLT) On a multiclass network without restricting the transmission to only one class per linkpertimeslot,Q isminimizedbytheminimumdelaypolicyofSec.4.3.2. 106 MulticlassMinimumDelayRouting Proof. Tosimplifytheproof,weassumearrivalsarei.i.d. overtimeslots,withtheunderstanding that it can easily be modified to yield similar result for non-i.i.d. arrivals, using the N-slot analysisderivedfrom (3.29). We often drop the timeslot variable(n) for ease of notation. Consider a stabilizable arrival rate o a ◦ interior to the stability region of a “generic” control policy that acts based only on current queue backlogs and current channel states. Also consider theG(f f, o q ◦ ,n) functional in(4.18)thatismaximizedbytheminimumdelaypolicyateachslotn. Ifthegenericpolicyalso maximizesGateachtimeslot,byAssum.4.1,itwillresultinthesametimeaverageperformance asthatoftheminimumdelaypolicy. Thus,weassumethattheGobtainedbythegenericpolicy isnotmaximal. Thisimpliestheexistenceofasufficientlysmallscalar ϵ> 0andsuchacontrol policy (possibly unfeasible) that stabilizes o a ◦ +ϵ 1 at each timeslot while makingG greater than or equal to that of the generic control policy. Let us refer to this latter control policy as “fictitious”aswedonotintendtoknowhowitreallyworks. Torestassurethatsuchapolicy alwaysexists,onemayendowitwiththeabilityofperfectprecognitionthatcanpreciselypredict thefuture arrivalsandchannelstates. Letf f(n)and o q ◦ (n)denotethetimeslotquantitiesprovidedbythegenericcontrolpolicyand letf f ′ (n)bethetimeslottransmissionsprovidedbythefictitiouspolicygiven o q ◦ (n). Extracting G from (4.26),G(f f ′ , o q ◦ ,n)>G(f f, o q ◦ ,n) leadsto 2(o a ◦ +ϵ 1) ⊤ o q ◦ +(o a ◦ +ϵ 1) ⊤ (o a ◦ +ϵ 1)− 2f f ′⊤ IB ◦ ⊤ (o a ◦ +ϵ 1)− ∆ W ′ > 2 o a ◦ ⊤ o q ◦ + o a ◦ ⊤ o a ◦ − 2f f ⊤ IB ◦ ⊤ o a ◦ − ∆ W which holds at each slot n. Taking expectation with respect to the randomness of arrivals, channelstatesandroutingdecision(incaseofarandomizedpolicy),weobtain 2ϵ 1 ⊤ o q ◦ +2ϵ 1 ⊤ o a ◦ +ϵ 2 1 ⊤ 1+2E{IB ◦ f f} ⊤ E{o a ◦ }+2Cov{IB ◦ f f, o a ◦ }+E{∆ W}> 2E{IB ◦ f f ′ } ⊤ E{o a ◦ }+2ϵ 1 ⊤ E{IB ◦ f f ′ }+2Cov{IB ◦ f f ′ , o a ◦ }+E{∆ W ′ } where for two random variablesX andY ,Cov{X,Y} := E{X ⊤ Y}− E{X} ⊤ E{Y} and Var{X} :=Cov{X,X}. Sincef produced by the generic policy is independent of arrival statistics, we getCov{IB ◦ f f, o a ◦ } = 0. We also haveE{IB ◦ f f ′ } = E{IB ◦ f f}+ϵ due to the feasibility condition (4.5) and the i.i.d. assumption on arrivals and channel states. Plugging theseidentitiesintheaboveinequalityyields 2ϵ 1 ⊤ E{o q ◦ }+E{∆ W}> 2ϵ 1 ⊤ E{IB ◦ f f ′ }− ϵ 2 1 ⊤ 1+2Cov{IB ◦ f f ′ , o a ◦ }+E{∆ W ′ }. 4.7SimulationResults 107 Summingovertimeslots0untilτ − 1,dividingthesumbyτ andtakingalimsupofτ →∞ frombothsidesleadtothefollowingexpectedtimeaverageinequality: 2ϵ 1 ⊤ o q ◦ > 2ϵ 1 ⊤ (IB ◦ f f ′ )+2Cov{IB ◦ f f ′ , o a ◦ }− ϵ 2 1 ⊤ 1 (4.31) whereweusedthefactthat limsup τ →∞ E{W(τ )− W(0)}/τ vanishesforanystabilizedtraffic matrixasW(n)= o q ◦ (n) ⊤ o q ◦ (n) isfinitewithprobability1ateachslot n. Nowconsidertheminimumdelaypolicyandsupposethatitprovidesf f ⋆ (n)and o q ⋆ ◦ (n)at eachslotn. Again,letf f ′ (n)beprovidedbythefictitiouspolicyateachslot n,butthistimegiven thecurrentqueuebacklogs o q ⋆ ◦ (n). DuetoTh.4.1,theinequalityG(f f ′ , o q ⋆ ◦ ,n)6G(f f ⋆ , o q ⋆ ◦ ,n) holdsateachslotn. Performingsimilarsteps,letusextractGfrom (4.26),takeexpectation,use E{IB ◦ f f ′ }=E{IB ◦ f f ⋆ }+ϵ ,sumovertimeslots0untilτ − 1,dividethesumbyτ andtakea limsupofτ →∞toobtain 2ϵ 1 ⊤ o q ⋆ ◦ 6 2ϵ 1 ⊤ (IB ◦ f f ′ )+2Cov{IB ◦ f f ′ , o a ◦ }− ϵ 2 1 ⊤ 1. (4.32) Notetheminimumdelaypolicyisthroughputoptimalandsostabilizesthearrivalrate o a ◦ ,which means limsup τ →∞ E{W ⋆ (τ )− W ⋆ (0)}/τ vanishes. Comparing (4.31) and (4.32) entails ϵ 1 ⊤ o q ⋆ ◦ 6 ϵ 1 ⊤ o q ◦ . Therefore, for any generic control policy which acts based only on current queue congestion and current channel states and is ableto stabilizeanarrivalrate o a ◦ viacreatingexpectedaverage queuelengths o q ◦ ,thereexists suchanϵ> 0asϵ 1 ⊤ o q ⋆ ◦ 6ϵ 1 ⊤ o q ◦ ,where o q ⋆ ◦ representstheexpectedaveragequeuelengthsifthe minimumdelaypolicyisexploitedtostabilize o a ◦ . Thenobservingthatϵ 1 ⊤ o q ⋆ ◦ 6ϵ 1 ⊤ o q ◦ isan equivalentexpressionforQ ⋆ 6Q concludestheproof. 4.7 SimulationResults Considerawirelessnetworkwith50nodesthatarerandomlydistributedonasquaresurface. We placealink betweeneverytwooffsetnodesthattheirproximitydistanceislessthanathreshold, which we considered it as 15% of the length of the surface. We then add the minimum number of extra required links with the minimum possible lengths to make the network connected. The result is a network of 128 links with the topology shown in Fig. 4.2. Wireless links are considered to be two-way channels, i.e., for any directed linkij∈E there existsji∈E with thesamecapacity. Thenetworkrunsunder1-hopinterferencemodelinwhichthelinkswith commonnodecannottransmitatthesametime. 108 MulticlassMinimumDelayRouting Everytimeslot,thecapacityofeachlinkij followsaGaussiandistributionwiththemeanm ij and the variance equal to 150. To assignm ij to different links, we adopt Shannon capacity with powertransmissionP ij ,noiseintensityN ij ,andabandwidthof1500,viz. m ij =1500log 2 (1+ P ij /N ij ). We randomly assign a noise intensity Nij ∈ [1,5] to each link at first and keep it fixedduringthesimulation. 4.7.1 Uniclasssensornetwork Inthissimulationweassumethateachnodeisasensorthatsendspacketstoauniquedestination as the sink which is indicated with a ring in Fig. 4.2. For this case of low power sensor network, let us take P ij = 2 for all links. We let exogenous packets arrive the network at each node, except for the sink, following Poisson distributions with parameterλ . We assume arrivals at differentnodesarei.i.d. overtimeslotsandwithrespecttoeachother. Evaluatingtheuniclasstransient performanceagainstBPpolicy,Fig.4.3displaystimeslot evolutionoftotalqueuecongestionfortwoPoissonparametersλ =1andλ =10,wherethe system starts from zero initial conditions, i.e, empty queue buffers. Noticeable is the small steady-state oscillations under the minimum delay policy contrary to large variations under BP policy thatconfirms the effectof (i) takinglink weights as quadraticin link queue-differential, (ii)schedulingwirelesslinksbasedonactualtransmittablepacketsratherthanlinkcapacities, Fig.4.2 Simulation testbench with 50 nodesrandomlydistributed onasurface. Linksareplaced betweeneverytwooffsetnodeswithaproximitydistancelessthanathresholdandextralinkswith minimum length are added to make the network connected, which result in 128 two-way wireless links. The node with a ring indicatesthesinkforthecaseofuniclasssensornetwork. 4.7SimulationResults 109 timeslot 0 2000 4000 6000 8000 10000 0 1 2 3 4 x 10 4 Total queue congestion 0 2000 4000 6000 8000 10000 0 1 2 3 4 5 6 x 10 4 Total queue congestion 0 400 800 1200 1600 2000 0 1 2 3 4 x 10 4 0 400 800 1200 1600 2000 0 1 2 3 4 5 6 x 10 4 4 10  Minimum delay policy BP policy BP policy Minimum delay policy 4 10  4 10  Minimum delay policy BP policy BP policy Minimum delay policy 4 10  timeslot 10000 10000 timeslot timeslot Fig. 4.3Timeslotevolutionoftotalnumberofpacketsintheuniclasssensornetwork,displayingtheperformanceofminimum delay policy versus BP policy for two arrival rates ofλ = 1 (left) andλ = 10 (right) at each node. The lower panels are zoomed in the 0-2000timeslotintervaltoemphasize the transient performanceof the twopolicies. 110 MulticlassMinimumDelayRouting and (iii) restricting packet forwarding to link queue-differential rather than spreading the most possiblenumberofpacketsalongeachactivatedlink. ComparingtheuniclassminimumdelaywithBPpolicy,Fig.4.4displaysaveragetotalqueue congestion, which is defined as the total number of packets queued in the network, as a function ofarrivalrate(Poissonparameterλ ). Eachbulletrepresentstheaveragetotalnumberofpackets in the sensor networkwhileλ increasesfrom 1 to10 inunit steps. The averagewas takenover the last40,000slots,where thesystemranfor50,000slotsstartingfromzeroinitialcondition. We dropped the first 10,000 slots to remove the transient effect on the steady-state condition. Theconnectingdashedlinesdisplay theseconddegreepolynomialinterpolation. Approaching the border of network capacity region, the two policies diverge even faster in delay performance. A crude explanation can be obtained by comparing the two control policies when q ij > µ ij for many links. One can see that the relative priority of links with higher queue differential increases faster in the minimum delay policy compared to BP, leading toanevenstrongerperformanceinheavytraffic. With q ij > µ ij , inforwarding, bothpolicies transmitµ ij number of packets along activated links. But in weighting, byq ij = µ ij +r for ar > 0, the minimum delay policy weights the links asw ij = µ 2 ij +2rµ ij , while BP does it asw ij =µ 2 ij +rµ ij . Thus,therelativepriorityoflinkswithhighqueuedifferentialincreases fasterintheminimumdelaypolicy,leadingtoamoreeffectivepacketdelivery. 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 x 10 4 Arrival rate at each node (packets/timeslot) Average total queue congestion (packets) 4 10  Minimum delay policy BP policy Fig. 4.4 Expected time average total queue length in the sensor network against the exogenous arrivalrates, changing fromλ = 1 toλ = 10,withthedashedlinesrepresentinginterpolation. 4.7SimulationResults 111 4.7.2 Multiclassmeshnetwork On the same simulation testbench of Fig. 4.2, we here assume that every node sends packets to every other node, forming a multiclass, multihop wireless network. Different classes are generatedateachnodefollowingPoissonrandomvariableswithparameterλ ,whereallofthem are i.i.d. over timeslots and with respect to each other. To support this multiclass traffic, we let each node expend 30 units of transmission power per timeslot, which under 1-hop interference modelleadstoP ij =30 foreachlink. Figure4.5displaystimeslotevolutionoftotalnumberofpacketsinthenetworkcontrolled by the minimum delay policy with and without SCLT strategy, compared with BP performance. Theresultsareshownforthreearrivalratesofλ =1,λ =5andλ =10packetsperclassper timeslot,wherethesystemstartsfromzeroinitialconditions,i.e,emptyqueuebuffers. Time average performance of the three control policies are compared in Fig. 4.6 with λ growing from 1 to 10 in unit steps. The average was taken over the last 40,000 slots, where the system ran for 50,000 slots starting from zero initial condition. Forλ = 1, average total numberofpacketsundertheminimumdelaypolicywithMCLTisonly220Kpackets,compared with15,400KpacketsunderBPpolicy,and9,380Kpacketsundertheminimumdelaypolicy subject to SCLT. Another important notice is that the difference in average network delay grows polynomiallywithrespecttothearrivalrateλ ,wheretheconnectingdashedlinesdisplaythe seconddegreepolynomialinterpolation. ComparedtoBP,theminimumdelaypolicy,eitherunderSCLTorwithMCLT,leadstoa superiorperformance. Nonetheless,theverynoticeablereductioninnetworkdelayhappenswith MCLT,wheretheroutingcontrolneedstocarefullydeterminethenumberofpacketsthatshould be incorporated from each class in each of scheduling and forwarding stages. Under SCLT, whichispracticedinBPschemes,ateachtimeslot,themulticlasssystemissomewhatsimplified tobeuniclassinthelinklevel. Atthesametime,foreachactivatedlink,BPschemessendas many packets as possible from one selected class consistently with the link capacity. These two characteristics of BP schemes in multiclass networks not only increase the average network delay dramatically, but also depletes the network resources by unnecessary packet forwardings. TheproposedminimumdelaypolicywithMCLTaddressesbothofthesedeficiencieswithout addinganymorecomplexityorrequirement. 112 MulticlassMinimumDelayRouting 0 2000 4000 6000 8000 10000 0 4 8 12 16 x 10 6 Total queue congestion 0 2000 4000 6000 8000 10000 0 0.5 1 1.5 2 2.5 x 10 7 Total queue congestion 0 2000 4000 6000 8000 10000 0 1 2 3 4 x 10 7 Total queue congestion 6 10  7 10  7 10  timeslot Total queue congestion (packets) Total queue congestion (packets) Total queue congestion (packets) Minimum delay policy with MCLT BP policy Minimum delay policy with MCLT Minimum delay policy with MCLT Minimum delay policy under SCLT Minimum delay policy under SCLT Minimum delay policy under SCLT BP policy BP policy Total queue congestion (packets) Total queue congestion (packets) Total queue congestion (packets) Fig.4.5Timeslotevolutionoftotalnumberofpacketsinthemulticlassnetworkforthreearrival ratesofλ = 1 (top),λ = 5 (middle) andλ = 10 (bottom)foreachclassateachnode. 4.8Conclusion 113 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 x 10 7 Arrival rate of each class at each node (packets/timeslot) Average total queue congestion (packets) 7 10  Minimum delay policy with MCLT BP policy Minimum delay policy under SCLT Fig. 4.6 Expected time average total queue length in the multiclass network against the exogenous arrival rates, changing fromλ = 1 toλ = 10,withthedashedlinesrepresentinginterpolation. 4.8 Conclusion Forstochasticmulticlasswirelessnetworkswithinter-channelinterferenceandtime-varying topology,wedevelopedadynamicroutingcontrolthatrequiresnoknowledgeofstatisticsand probabilitiesofrandomvariablesinthesystem. Besidesthroughputoptimality,itminimizes average network delay within the class of all routing algorithms that perform based only on current queue congestion and current channel states. This important class includes all op- portunistic max-weight schedulers that do not incorporate the Markov structure of topology process into their decisions, among which is Back-Pressure (BP) and most of its derivations. It also includes all stationary randomized algorithms that make a routing decision as a pure (possiblyrandomized)functiononlyofcurrentchannelstates,whichtypicallyrequiretheperfect probabilityknowledgeofarrivals andchannelstates. Beyond wireless networks, our proposed minimum delay policy is a solution to the general problemofservicedelayminimizationforasetof interdependentandtime-varying resources thatservearandomlyarrivingcustomerswithrandomservice-time. Chapter5 MulticlassMinimumCostRouting Minimum cost routing is considered at network layer of multiclass multihop wireless networks influenced by stochastic arrivals, inter-channel interference and time-varying topology. Endow- ing each wireless link with a cost factor, possibly time-varying and different for each class, we propose a routing control that minimizes the sum of square of link packet transmissions weighted by the link cost factors. ThistotalquadraticcostisnamedtheDirichletroutingcostas itcloselycorrelateswiththeDirichletenergyinheatcalculus. Bymappingwirelessnetwork into a nonlinear resistive network, we first show that the fluid limit of packet transmissions complieswithageneralizationofOhm’slawinamulticlassfashion. Routingcostminimization thenreliesonDirichlet’sprinciple. Amongotherfeaturesoftheproposedcontrolpolicy,itis throughputoptimalandrequiresnoinformationaboutnetworktopologyandpacketarrivals. 5.1 Introduction Consideramulticlassslotted-timestochasticwirelessnetworkinwhichnewpacketsofthesame size may randomly arrive at different nodes, destined for any other node, potentially several hopsaway,where(i)duetoenvironmentalfactorsandusermobility,topologyofthenetwork mayrandomlychangeintime;and(ii)duetointer-channelinterference,unlikewirelinelinks thatareindependentresources,notallwirelesslinkscantransmitatthesametime. Wedescribethenetworkbyasimple,directedconnectivitygraphwithsetofnodesV and directed edgesE. Packets of the same destination form a class (regardless of their sources), whereK⊆V representsthesetofallexistingclassesinthenetwork. Eachnodei∈V holdsa separatequeueq (d) i foreach d-classtotransmitoveritsoutgoinglinks. 116 MulticlassMinimumCostRouting Given a wireless linkij∈E and a classd∈K, the link actual-transmissionf (d) ij (n) counts the number ofd-class packets (ord-packets in short) transmitted over the link at timeslotn. For eachclassd,everylinkij isalsoendowedwithalinkcostfactorρ (d) ij (n)>1thatrepresentsthe costoftransmittingoned-packetoverthelinkatslotn. Weaimatfindingaconfigurationof timeslotactual-transmissionsf (d) ij (n) thatsolvestheoptimizationproblem min limsup τ →∞ 1 τ τ − 1 X n=0 E n X ij∈E X d∈K ρ (d) ij (n) f (d) ij (n) 2 o s.t. Queue Stability for all Stabilizable Arrivals (5.1) whereEdenotesexpectation. Werefertothequadraticcostfunctionofproblem (5.1)asthe Dirichlet routing cost due to its close correlation with Dirichlet’s principle in heat calculus [90], whichshallbeexplainedinthischapter. Worthtonotethattheminimizationproblem (5.1)needstobesolvedatnetworkandlink layer,whichtotallydifferentiatesthetreatmentofthisproblemfromcross-layeroptimization techniques[66–69]. Theaiminthelatteristocontroltheflowofpacketsattransportlayersoas to keep the arrival rates into network layer within the network capacity region. A network-layer routingpolicy,ontheotherhand,hasnocontrolonarrivals;instead,thebasicassumptionhereis queue stabilizability, meaning that the arrival rates lie within network capacity region. Then, an important quality factor of a routing protocol, called throughput optimality, is to stably support the entirecapacity region. Itshould beobvious thatminimum costrouting atnetworkand link layer neither has a contradiction with, nor is a substitution for the flow control at transport layer thatseekstokeepthenetworklayerstabilizable. 5.1.1 DiscussionontheDirichletroutingcost The quadratic form of cost function in (5.1), i.e., the total weighted square of the number of d-packettransmissionsoverlinks,isdecreedbyDirichlet’sprincipleuponwhichourproofof optimality is founded. As we will show, this mimics the concept of power loss in a resistor, defined by the square of electrical current weighted by the resistance. One might be more interested in defining a minimization target that is linear in number of transmitted packets, consisting of a sum of costs multiplied by the number of packets suffering those costs. Then we cannolongerclaimthatourprotocolstrictlyminimizessuchalinearroutingcost,byexactlythe same way that the power dissipation on a resistive network would not necessarily be minimized ifitweredefinedaslinearinelectrical current. 5.1Introduction 117 Whenlinkcostfactorsarecorrelatedwithasortofresourceconsumptionsuchasenergy (resp., quality defect such as end-to-end delay), the constrained optimization problem (5.1) becomescloselyrelatedtominimizingaverageresourceusage(resp.,averagequalitydefects) subjecttothroughputoptimality. Suchaminimizationtargetcanbedefinedinvariouswaysthat representsdifferentroutingpenalties,givingawide-reachingimpacttoproblem (5.1). Below aresomeexamplesthatcanbebroughtintothisabstractmodel. Link quality factor turned as a cost factor: The quality of wireless communication depends onhardwareandenvironmentalfactorssuchasbandwidth,energy,noise,fadingandobstruction, among others. In literature, different state-of-the-art link-quality metrics have been introduced such as Expected Transmission Count (ETX), Packet Reception Rate (PRR), Required Number of Packet Transmissions (RNP), Expected Transmission Time (ETT) and Effective Number ofTransmissions(ENT).Acomprehensivereviewonlinkqualityestimatorscanbefoundin [65, 91, 92]. Most link quality measures can directly merge into problem (5.1) as link cost factorswithminimummodificationsrequired. Routing distance minimization: The cost factor of each linkij for a classd can be assigned proportionaltothehop-countorgeographicaldistancebetweenthereceivingnodej andthe final destination node d. These distances can either be estimated or computed by running a shortest path algorithm. Then, due to the close correlation between routing distance and packet latency,problem (5.1)canindeedbeinterpretedasaverageend-to-enddelayminimization. Energy usage minimization: Wireless link capacity is a function of transmission power and channel condition. Assuming that power allocation happens independently of queue congestion, eachlinkmayreceiveacostfactorproportionaltotheratioofallocatedpowertoitscapacity. Thisway,problem (5.1)canbeexplained as theminimization of average energyconsumption through thenetworksubjecttoqueuestability. Relayingcostminimization: Nodesinad-hocnetworks,whichbelongtodifferentusers,may act in their own interests rather than forwarding traffic for others. For example, a node incurs an energycost(andpossiblyamemory/computationcost)whencontributingtopacketrelaying. Severalprotocolshaveaddressedthisnon-cooperativebehavior,basedmostlyongametheory and with no cost optimization or throughput optimality results (see [93] and references therein). Assume that every node declares a relaying cost for transferring one packet. Then to minimize total routing cost in the framework of problem (5.1), each incoming link to a node can receive a costfactorproportionaltothenoderelayingcost. 118 MulticlassMinimumCostRouting 5.1.2 Relatedworks It was shown in [63] that a stationary randomized algorithm can solve the constrained opti- mizationproblem (5.1). Whilesuchanalgorithmexistsintheory,itisintractableinpractice, as it requires a full knowledge of arrival statistics and channel state probabilities. Moreover, assumingallthestatisticsandprobabilitiescouldbeaccuratelyestimated,thealgorithmstill needstosolvea dynamic programmingproblemforeachtopologystate,wherethenumberof statesgrowsexponentiallywiththenumberofwirelesschannels. Thus far, V-parameter Back-Pressure (BP) of [63] has been the only feasible approach to decreasing—not minimizing—a generic routing penalty at network layer. The original BP was first described by [ 44] in the context of wireless networks and independently discovered later by [94] as a low-complexity solution to certain multi-commodity flow problems. The V-parameter BP adds a term with constantV> 0 to the original BP [44] that trades queue congestion for routingcost. ItisproventhattheaverageroutingcostcancomewithinO(1/V)ofitsminimum, but at the expense of increasing average network delay byO(V) relative to that of the classical BP.Specifically, whilethealgorithmcan decrease a routing cost, it is not able toachieve the minimumroutingcostsubjecttonetworkstability. Weremark,however,thatV-parameterisnot restrictedtothequadraticstructureoftheDirichletroutingcostin (5.1). Addingdistanceinformationtolinkweights,[95]enhancesBPtogiveprioritytoshorter paths. However, it does not minimize average routing distance and needs shortest path informa- tion. Energy-delaytradeoffinsensornetworkswasconsideredin[ 96]. Asymptoticenergyusage as network size grows to infinity was studied in [ 97]. An adoption of V-parameter BP was used in[98]toreduceaverageenergyusageinamultihopwirelessnetwork. Abetterenergy-delay tradeoff was introduced in [ 99] for the special case of wireless downlinks, but not for multihop wirelessnetworks. Todecreaseaveragepackettransmissions,[56]integratedETXmetricinto V-parameter BP. Assuming each node has a-priori information of its hop distance from all othernodes,[100]restrictedBPtothenumberofhop-countsoneachflowtoreducenetwork delay. Using greedy embedding, [49] modified BP by giving priority to the links with a shorter hyperbolicdistancetothefinaldestinationtoimproveBPlatencyinlighttraffic. In the context of making analogy to electrical networks, the classic early work in [101] modeledamulti-commodityflowbyacollectionofelectricalnetworksandusedcurrent-voltage pairs to find optimal primal-dual decision variables. Later on, [ 102] introduced active elements suchasoperationalamplifierstorealizedifferentidealizedelectricalcomponentsinsolvinga broaderrangeofnonlinearoptimizationproblems. 5.1Introduction 119 5.1.3 Contribution This is the first time in literature that such a generic multiclass routing penalty as the one in problem (5.1) can be minimized at networkand linklayersubjecttoqueuestabilityandwithout requiringanyinformationaboutnetworktopologyandtraffic. Infact,weextendtheresultsof minimum routing cost under Heat-Diffusion policy, which was developed for uniclass networks inCh.3,intwochallengingdirectionsasfollows: First,asopposedtotakinganuncapacitateddirectedgraph,hereweformalizeDirichlet’s principleonacapacitateddirectedgraphasthegenuinerepresentationofapacketnetwork. This overcomesthemainrestrictioninCh.3,thatedgecapacitiesdonotinfluenceheatflowequations. Second,hereweconsidermulticlassnetworksthattogetherwithlimitededgecapacitiesmakethe extensionofDirichlet’sprinciplemoreunorthodoxanditsanalyticalprooffarmorecomplicated. This nontrivial extension is made possible, first, by developing a novel concept of multi- charge electric conduction on a nonlinear resistive network with capacitated directed edges, and second, by showing that under our proposed routing protocol, the long-term average dynamics (fluid limits) of a multiclass wireless network comply with multi-charge nonlinear Ohm law. Bringing stochastic wireless and deterministic electrical networks together can open a new way totakeadvantageof classictools fromcircuittheory, suchaseffectiveresistance,intheanalysis andoptimizationofstochasticprocessingnetworksconstrainedtolinkinterferenceandcapacity. Further, when all wireless channels are of unit cost factor for all classes, our multiclass protocolherereducestotheonedevelopedinCh.4. Thisleads,besidesroutingcostminimization, tominimumaveragenetworkdelayamongallroutingprotocolsthatactbasedonlyoncurrent queuecongestionandcurrentchannelstates. Our results on minimum routing cost here along with the results on minimum routing delay in Ch. 4 make the pillars to extend the Pareto optimal performance of Ch. 3 to multiclass processing networks and without restriction that the edge capacities do not intervene in the physicsofdiffusiveconduction. Last but not least, our proposed protocol enjoys the same algorithmic structure, complexity, and overhead signaling as BP. Thus, all advanced improvements to BP (see [63] and references therein)caneasilybeleveragedtofurtherenhancethenewprotocolaswell,whichalsosimplifies thesoftwaretransitiontopracticefromexistingBPimplementations. 120 MulticlassMinimumCostRouting 5.1.4 Organization Next section provides a prelim on multiclass multihop wireless networks including state model of the system and multiclass V-parameter BP algorithm. In Sec. 5.3 we generalize Ohm’s law and Dirichlet principle to nonlinear resistive networks subject to edge directionality and capacity. Inyetanothergeneralization,Sec.5.4extendstheresultstomulti-chargenonlinear networks. Inspired by the multi-charge conduction, we introduce a Dirichlet-based routing policy in Sec. 5.5. Analytical proof of routing cost minimization lie in Sec. 5.6 followed by the proof of throughput optimality in Sec. 5.7. Simulation results are discussed in Sec. 5.8. The chapterisconcludedbySec.5.9whichalsodiscussessomefutureresearchproblems. 5.1.5 Notation WeusethesamenotationasinCh.3(seeSec.3.1.4). Todenotemulticlasshyper-arrays,weuse blackboard bold typeface, such asf f, where certain lines of the symbol are doubled. ForA as a matrix,vec(A) transformsittoacolumnvector. 5.2 Preliminaries Consider a multiclass multihop wireless network in which new packets randomly arrive at differentnodesandhavedifferentdestinations,wherepacketsofthesamedestinationforma class. Todirectpacketsfromtheirsourcestowardtheirdestinations,aroutingprotocolmakes routingdecisionsateverytimeslotthatconsistsofactivatingasetofwirelesslinks,selecting whichclass(es)tobeservedbyeachactivatedlink,andassigningthenumberofpacketstobe sentoneachactivatedlink. Weassumethatpacketsarenotsenttotrappingnodes,i.e.,ifanode acceptsd-packets, there must exist at least one possible route from that node to the destination noded. Though this assumption is not necessary for any of our analytical results, it ensures thatadynamicrouting,withnorouting pathconstraint, willnotmistakenlysenda packettoa trappingnodethatpreventsitfromeverreachingitsdestination. 5.2.1 Stabilityandthroughputoptimality Adiscrete-timestochasticprocessx(n) iscalledstableif x:=limsup τ →∞ 1/τ X τ − 1 n=0 E{x(n)}<∞ (5.2) 5.2 Preliminaries 121 withEdenotingexpectation. Thedefinitionofstabilityandtheoverbarnotationareextended entrywise to vectors and matrices. A network is called stable if all its queues are stable. A traffic matrix, which represents the arrival rate of different classes into different nodes, is called stabilizable if there exists at least one routing policy (possibly unfeasible) that can stabilize the networkunderthattraffic. Foraroutingpolicy, stabilityregion isthesetofall trafficmatrices thatitcanstablysupport. Networkcapacityregionistheunionofthestabilityregionsofall possible routing policies (possibly unfeasible). A routing policy is called throughput-optimal if its stability region coincideswith the network capacity region; thusit securesnetwork stability forallstabilizabletrafficmatrices. 5.2.2 Inter-channelinterference Unlike wireline networks where links are independent resources, two wireless links cannot transmitif they get interference. Aninter-channelinterferencemodelspecifiestheserestrictions onsimultaneous transmissions. Givenaninterferencemodel, wedefinea maximalschedule as asetoflinkssuchthatnotwolinksinterferewitheachother,andnomorelinkcanbeadded toitwithoutviolatingtheconstraintsofinterferencemodel. Wedescribeamaximalschedule withaschedulingvectorπ ∈{0,1} |E| whereπ ij takesthevalue1ifthechannelij isincludedin themaximal schedule, and0 otherwise. Givenaconnectivitygraph(V,E),wealsodefinethe schedulingsetΠ asthecollectionofallmaximalschedulingvectors. Observethattheschedulingsetvariesaccordingtointerferencemodel. Ourresultsarevalid forthecategoryofallinterferencemodelsinwhichanodecannottransmitpacketstomorethan one neighbor at each timeslot, i.e., a node may receive packets from several of its incoming linksandatthesametimemaytransmitpacketsoveroneofitsoutgoinglinks. Tothebestof our knowledge, interference constraints in all current network layer protocols, including general K-hopinterferencemodels,fallinthiscategory. 5.2.3 Time-varyingtopology Networktopologymayvaryintimeduetonodemobilityandsurroundingconditionssuchas obstacle effect and channel fading. We assume that the sets V andE change far slower than channel states and so we can take them fixed during the time of interest. Then a temporarily unavailable channel is characterized by a zero link capacity and persistent variations are caught by updating the connectivity graph(V,E) in a long scale regime. We also assume that channel 122 MulticlassMinimumCostRouting states remain fixed during a timeslot, while they may change across slots according to some (unknown)probabilitylaws. Let a stochastic processS(n) = S 1 (n),··· ,S |E| (n) represent channel states at slot n, whichdescribesalluncontrollableconditionsthataffectchannelcapacities,andpossiblylink cost factors. Assume thatS(n) evolves according to an ergodic stationary process and takes values ina finite(but arbitrarilylarge) set S. For example,an irreducible Markov chain orany i.i.d. sequenceofstochasticmatricesarebothergodicandstationary. ThenduetoBirkhoff’s ergodictheorem,eachstateS∈S hasaprobability s:=P S(n)=S =limsup τ →∞ 1/τ X τ − 1 n=0 I S(n)=S (5.3) where P S∈S s = 1 andI X is the indicator function. Though our routing protocol does not requirethestateprobabilitiess,theexistenceofsisimportantforestablishingnetworkcapacity region,andalsotheoreticalanalysisofourprotocol. Specifically,itensurestheexistenceof E µ (n)}= X S∈S sE µ (n) S(n)=S E ρ (n)}= X S∈S sE ρ (n) S(n)=S (5.4) whereµ (n)andρ (n)representthevectorscomposedofchannelcapacitiesµ ij (n)andlinkcost factorsρ ij (n),respectively,atslotn. 5.2.4 Statemodelofmulticlassnetworks Letq (d) i (n)representthenumberofd-packetsinthenodeiatslotn. Thehyper-vectorofstate variablesformulticlassqueuingsystemisobtainedas o q ◦ (n):=vec q (1) ◦ (n),...,q (|K|) ◦ (n) ∈R (|V|− 1)|K| q (d) ◦ (n):= q (d) 1 (n),...,q (d) d− 1 (n),q (d) d+1 (n),...,q (d) |V| (n) ⊤ withq (d) d (n)≡ 0 discarded from the set of states. Note that packet leaves the network as soon as reachingitsdestination,andsothebacklogofd-packetsattheirdestinationnoded iszero. Notation 5.1. We use subscript◦ along with superscript(d) to denote reduced arrays obtained by discarding the entries corresponding to the destination noded. Subscript◦ on multiclass 5.2 Preliminaries 123 symbols,whichisnotaccompaniedbysuperscript(d)anymore,denotesreducedhyper-arrays obtainedbyconcatenatingallcorrespondingreducedarraysofdifferentclasses. Let the stochastic processa (d) i (n) represent the number of exogenousd-packets arriving into thenodei atslotn. Weobtainthehyper-vectorofnodearrivals o a ◦ (n):=vec a (1) ◦ (n),...,a (|K|) ◦ (n) ∈R (|V|− 1)|K| a (d) ◦ (n):= a (d) 1 (n),...,a (d) d− 1 (n),a (d) d+1 (n),...,a (d) |V| (n) ⊤ witha (d) d (n)≡ 0 discardedfromthesetofarrivals. Foreachlinkij∈E,itscapacityµ ij (n),whichisfrequentlycalledlinktransmissionrate inliterature,countsthemaximumnumberofpacketsthelinkcantransmitatslotn. Thelink actual-transmissionf (d) ij (n),ontheotherhand,countsthenumberofd-packetsgenuinelysent overthelinkatslotn. Whilelinkcapacitiesvaryonlybychannelstates,linkactualtransmissions aredeterminedbyaroutingprotocolsubjectto 06f (d) ij (n)6min{q (d) i (n), µ ij (n)}. Letuscomposethehyper-vectoroflinkactual-transmissionsas f f(n):=vec f (1) (n),...,f (|K|) (n) ∈R |E||K| f (d) (n):= f (d) 1 (n),...,f (d) |E| (n) ⊤ . RecallfromSec.3.4thatgivenadirectedgraph(V,E),wedenoteitsnode-edgeincidence matrixbyB in whichB iℓ is1if nodei isthetailofdirectededgeℓ,is− 1 ifiisthehead,and is0otherwise. Foraclassd,considerthereducedincidencematrixB (d) ◦ obtainedfromB by discarding the row related to the destination noded. We refer toB (d) ◦ as the basis incidence matrix with respect to noded, or classd for thatmatter. Concatenatingbasis incidencematrices ofdifferentclasses,themulticlassbasisincidencematrixisobtainedas IB ◦ :=diag B (1) ◦ ,...,B (|K|) ◦ ∈R (|V|− 1)|K|×|E||K| . InheritedfromB ◦ ,matrixIB ◦ hasalsofullrowrank. OnecanverifythatIB ◦ f f(n) isahyper- vectorinwhichtheentrycorrespondingtonodei andclassd is (IB ◦ f f) (d) i (n)= X b∈out(i) f (d) ib (n)− X a∈in(i) f (d) ai (n). 124 MulticlassMinimumCostRouting Puttingthesepiecestogether,weobtainatimeslotrepresentativeofthef f-controlledstochas- ticstatedynamicsforthemulticlassqueuingsystemas o q ◦ (n+1)= o q ◦ (n)+ o a ◦ (n)− IB ◦ f f(n). (5.5) 5.2.5 MulticlassV-parameterBPpolicy In the original Back-Pressure (BP) [44], at each timeslot, each link receives a weight as the product of its capacity and the queue differential of its optimal class, which is the class with maximumlinkqueue-differential. Thenasetofnon-interferinglinkswithmaximumcumulative weightarescheduledforpacketforwarding. To incorporateacost functioninto thealgorithm, V-parameter BP [63] penalizes each link with its related cost via a user-assigned parameter V ∈ [0,∞) that determines the worthiness of reducing the cost at the expense of increasing networkdelay,whereV =0 recovers theclassicalBP. To treat the Dirichlet routing cost (5.1) as a penalty function, V-parameter BP observes queuebacklogsq (d) i (n)andestimateschannelcapacitiesµ ij (n)andlinkcostfactorsρ (d) ij (n)at eachtimeslotn tomakearoutingdecisionasfollows. 1) MulticlassV-parameterBPweighting: Oneverydirectedlinkij andforeachclassdfind thelinkqueue-differential q (d) ij (n):=q (d) i (n)− q (d) j (n) andselecttheoptimalclass d ∗ ij (n):=argmax d∈K q (d) ij (n). (5.6) Thengiveaweighttothelinkusingitsestimatedcapacityas w ij (n):=µ ij (n) q (d ∗ ) ij (n)− Vρ (d) ij (n)µ ij (n) + . (5.7) 2) MulticlassV-parameterBPscheduling: Findaschedulingvectorsuchthat5 π (n)=argmax π ∈Π X ij∈E π ij w ij (n) (5.8) wheretiesarebrokenrandomly. 3) MulticlassV-parameterBPforwarding: Overeachactivatedlinkij withw ij (n)>0transmit packets from the classd ∗ ij (n) at full capacityµ ij (n). If there is no enough packets from the classd ∗ ij (n) atnodei,transmitnullpackets. 5.3NonlinearConduction 125 5.3 NonlinearConduction Theideaofsolvingproblem (5.1)hasits rootindissipative powerminimizationthatnaturally occurs in the process of electrical conduction over a conducting medium and is mathematically explained by Dirichlet’s principle. As the first step to adopt this idea for multiclass wireless networks,thissectionenvisionsaso-calledmulti-chargenonlinearresistivenetworktoextend theconceptofDirichlet’sprincipleanddissipativepowerminimization. Inspiredbythis,the next section proposes a multiclass routing protocol that solves problem (5.1) by composing suchalong-termaverageflowofpacketsonwirelessnetworkthattakestheformofelectrical currentsonitsunderlyingnonlinearresistivenetwork. 5.3.1 Conductiononmanifolds ConsiderMasamanifoldofconductingmediumwithboundary∂M. LetA(z)beaspatial currentsourcedensityrelativetoavolumeformdv overM,andQ(z)betheinducedvoltage potential overM, whichis prescribed on the boundary∂M. LetF(z) be the vector field of electrical current at pointz∈M. By Gauss’s theorem, the outward flux through a bounded regionM ′ ⊂M isequaltothevolumeintegralofthedivergenceoverM ′ ,whichreads Z ∂M ′ F(z),ℵ(z) ds= Z M ′ divF(z)dv whereℵ(z)istheexteriornormal,dsisthemeasureoftheboundaryand⟨· ,·⟩ denotestheinner product. Inthesteady-stateequilibratedconduction,theprincipleofchargeconservation,on the other hand, asserts that the total charge entering into the bounded regionM ′ is equal to the chargeleavingouttheboundary,whichreads Z M ′ A(z)dv = Z ∂M ′ F(z),ℵ(z) ds. Combiningthesetwoequalitiestogether,weobtain Z M ′ divF(z)dv = Z M ′ A(z)dv. SinceM ′ isanarbitrarybounded region,onecanchooseitinfinitesimallysmall,leadingto divF(z)=A(z). (5.9) 126 MulticlassMinimumCostRouting Ohm’slawstatesthattheelectricalcurrentbetweentwopointsisproportionaltothegradient ofvoltageacrossthepointsscaledby theconductivity ofmaterial,whichreads F(z)=− σ (z)∇Q(z) (5.10) wheretheconductivityσ (z)isasymmetricpositivedefinitematrix. SubstitutingOhm’slaw (5.10)intoGauss’stheorem (5.9)leadsto div σ (z)∇Q(z) +A(z)=0 (5.11) whichisPoisson’sequationinacoordinate-freeformat. Dirichlet’s principle states that the Poisson’s equation (5.11) has a unique solution that minimizestheDirichletenergy E D Q(z) := Z M 1 2 σ ∥∇Q(z)∥ 2 − Q(z)A(z) dv amongalltwicedifferentiablefunctions Q(z)thatrespecttheprescribedvoltagepotentialon theboundary∂M [90]. 5.3.2 Conductiononresistivenetworks Rather than the smooth conducting manifoldM, now consider a resistive network(V,E) in which twoneighboring nodes areconnected via alinear lumped resistor. Exogenous electrical currents are injected into (resp., drawn from) the network via positive (resp., negative) current sources attached to different nodes. Let us fix zero voltage at a single reference node, also referredtoasground. Thissomewhatmimicscollapsingtheboundary∂Mtoapointonthe continuous domain. In particular, ground can be visualized as a node that collects the net electrical current injected into the network, i.e., algebraic sum of current sources, and drains it backintothesources,soastobuildaclosed flowsystem. To solve circuit problems, it is essential to assign an arbitrary algebraic-topological ori- entation to each edge with the understanding that the particular choice of orientation has no impact on the solutions. Accordingly, every edge variable is signed, while a negative quantity is interpreted to be on the opposite direction of the edge orientation. Using the notion of arbitrary edgeorientationonundirectedgraphs,onecanadoptthenode-edgeincidentmatrixB inthe samewayitwaspreviouslydefinedondirectedgraphs. 5.3NonlinearConduction 127 Letq representthevectorofvoltagesatthenodes,f thevectorofelectricalcurrentsthrough the edges,σ the vectorof edgeconductances, anda the vector of currentsources injected into the nodes. Withd being grounded, we assumea d ≡ 0 and define the reduced arrays q ◦ ,a ◦ and B ◦ bydiscardingtheentriesrelatedtothenoded. Thentheprincipleofchargeconservation (5.9)istranslatedtoKirchhoff’sCurrentLaw(KCL)onelectricalnetworks,whichreads B ◦ f =a ◦ . (5.12) Itassertsthatateachnon-groundednode,thealgebraicsumofcurrentsiszero. Observethat entriesofthenodevectorB ◦ f readthenetoutflowcurrentthroughtheresistiveedgesconnected toeachnode. Ohm’slaw(5.10),ontheotherhand,takestheformof f =diag(σ )B ◦ ⊤ q ◦ . (5.13) Then the combinatorial Poisson equation is obtained by substituting Ohm’s law (5.13) into Kirchhoff’scurrentlaw(5.12),whichreads − L ◦ q ◦ +a ◦ =0 with L ◦ :=B ◦ diag(σ )B ◦ ⊤ . (5.14) Observe that entries of the edge vectorB ◦ ⊤ q ◦ read the voltage difference across each edge. The linear operatorL ◦ is called the Dirichlet Laplacian with respect to ground and, as shown in Sec.2.3,ispositivedefinite foranyconnectednetwork. Like the classical case, combinatorial Poisson equation (5.14) has a unique solution that minimizesthecombinatorialDirichletenergy E D (q ◦ ):= 1 2 q ◦ ⊤ L ◦ q ◦ − q ◦ ⊤ a ◦ . (5.15) The proof is far simpler in the combinatorial case versus the smooth case. In fact, asL ◦ is positive definite, E D (q ◦ ) is convex. Thus, it reaches its minimum at the critical point where its first-ordervariationvanishes,whichreadilyleadstothePoissonequation (5.14). The celebrated implication of combinatorial Dirichlet principle is Thomson’s principle which states that the power dissipation on a linear resistive network is minimized under Ohm’s law. Foravectorofelectricalcurrentsf,thetotaldissipativeenergyisdefinedby E R (f):=f ⊤ diag(σ ) − 1 f. (5.16) 128 MulticlassMinimumCostRouting − Τ − Τ Τ − Τ Τ Τ 1 Fig.5.1Acapacitateddirectededgeismodeledbyanonlinearresistorinserieswithanidealdiode: (left)Current-voltagecurveofthenonlinearresistorwhichsaturatescurrentatµ ij andshowslinear conductivityofσ ij priortosaturation. (middle)Resistive-voltagecurveofthenonlinearresistor. (right)Current-voltage curve of the nonlinearresistorinserieswiththeidealdiode. Letf ∗ be the configuration of currents that minimizes E R (f) subject to KCL (5.12) and letq ∗ betheconfigurationofvoltagesthatminimizes E D (q ◦ )in (5.15). Onecanverifythatf ∗ and q ∗ arerelatedtoeachotherbyOhm’s lawin (5.13). 5.3.3 Conductiononnonlinearresistivenetworks Instead of undirected graph with capacity-free edges, now consider a graph under both edge directionality and edge capacity constraints. The corresponding resistive network will have the same configuration as the linear one with the exception that here, rather than using a linear resistor, we connect two neighboring nodes by a nonlinear resistor in series with an ideal diode. The diodes allow current to flow only along edge directions, while the nonlinear resistors limit electricalcurrenttoedgecapacities. Figure5.1providesagraphicaldescriptiononmimickingcapacitateddirectededgeswith a series of nonlinear resistors and diodes. Consider an edge ij with capacity µ ij and let σ ij represent the edge conductance in linear regime when the current is still below the limited edge capacity. The nonlinearresistor will beafunctionof voltagedifferenceacrosstheresistorand independentofedgedirectionality,whichisdescribedby r ij =    1/σ ij if|q ij |6µ ij /σ ij |q ij |/µ ij if|q ij |>µ ij /σ ij . Whenµ ij →∞,itrecoversr ij =1/σ ij foracapacity-freelinearresistiveedge. On any electrical network, KCL (5.12) remains unchanged. However, on our nonlinear resistive network, the Ohm’s law (5.13) must be modified to allow electrical current in only one direction, and to limit it within the edge capacity. Let the arbitrarily-chosen algebraic- 5.3NonlinearConduction 129 topological edge orientations concur with the edge directions. Then withµ being the vector of edgecapacities,thenonlinearOhmlawreads f =min diag(σ ) B ◦ ⊤ q ◦ + , µ . (5.17) Plugging the nonlinear Ohm law (5.17) into Kirchhoff’s current law (5.12), we obtain the followingnonlinearPoissonequationoncapacitateddirectednetworks: − ⃗ L ◦ (q ◦ )+a ◦ =0 ⃗ L ◦ (q ◦ ):=B ◦ min diag(σ ) B ◦ ⊤ q ◦ + , µ . (5.18) Wereferto ⃗ L ◦ (·) asthe nonlinearDirichletLaplacian. Givenaconnectedlinearresistivenetworkunderfinitecurrentsources,thecombinatorial Poisson equation (5.14) always leads to finite voltages at the nodes. However, for the nonlinear Poisson equation (5.18) to have a finite solution, two conditions need to be met: First, each nonzerocurrentsourcemustconnecttogroundthroughatleastonedirectedpath,wherethe pathdirectsfromthesourcetogroundforapositiveandviceversaforanegativecurrentsource. Second,theremustexistatleastoneconfigurationofedgecurrentsthatcantransfercurrents from all sources to ground while respecting alledge capacities. In theabsence of either of these twoconditions,thenetworkflowproblemhasindeednosolutioninthesensethatthereisno way to transfer all commodities to the destination, which in our case means to transfer electrical chargefromallcurrentsourcestoground. Observethatthefirstconditionisenforcedbythe edgedirectionality,whilethesecondconditionistheresultofedgecapacity. Definition 5.1. Avectorofcurrentsourcesisfeasibleifthereexistsatleastoneconfiguration ofedge currentsthatmutuallysatisfyKCLateachnodeandcapacityoneachedge. In the absence of edge capacity constraints, the above definition reduces to Def. 3.10 on capacity-freedirectedgraphs,whichonlyrequireseachcurrentsourcebeingconnectedtoground through atleastonedirectedpath. Contrary to the classical combinatorial LaplacianL ◦ on linear resistive networks, ⃗ L ◦ is an operand-dependent operator that retains neither linearity nor symmetry. Thus, the easy way of proving Dirichlet’s principle on uncapacitated undirected graphs ceases to exist here, as we can nolongerclaimthat ⃗ L ◦ (q ◦ )isthedirectionalderivativeof 1 2 q ◦ ⊤ ⃗ L ◦ (q ◦ )alongq ◦ . Nonetheless, bythenexttheoremweextendDirichlet’sprincipletocapacitateddirectedgraphs. 130 MulticlassMinimumCostRouting Theorem 5.1. (Nonlinear Dirichlet Principle) Consider a nonlinear resistive network with capacitated directed edges. Under a vector of feasible current sourcesa ◦ , the nonlinear Poisson equation (5.18)hasauniquesolutionthatminimizesthenonlinearDirichletenergy ⃗ E D (q ◦ ):= 1 2 q ◦ ⊤ ⃗ L ◦ (q ◦ )− q ◦ ⊤ a ◦ . (5.19) Proof. Using(5.18),onecanexpand ⃗ L ◦ toobtain ⃗ E D (q ◦ )= 1 2 q ◦ ⊤ B ◦ + min diag(σ ) B ◦ ⊤ q ◦ + , µ − q ◦ ⊤ a ◦ (5.20) where each entry ofB ◦ ⊤ q ◦ ∈ R |E| represents the voltage-difference along the corresponding edge. Letq ∗ ◦ bethe ⃗ E D minimizingsolution. We rearrangeand partitionB ◦ ⊤ q ∗ ◦ into positive, zero and negative components. Accordingly, the incidence matrix B ◦ gets partitioned as B ◦ = B ⊕ |B ∅ |B ⊖ , whereB ⊕ ,B ∅ andB ⊖ respectively contain the incidence information of edges with positive, zero and negative values in B ◦ ⊤ q ∗ ◦ . Likewise, the edge weights get partitionedintoσ = σ ⊕ |σ ∅ |σ ⊖ . Then ⃗ E D (q ◦ ) atq ◦ =q ∗ ◦ canbeshownas ⃗ E D (q ∗ ◦ )=− q ∗⊤ ◦ a ◦ + 1 2 q ∗⊤ ◦ B ⊕ + min diag(σ ⊕ ) B ⊤ ⊕ q ∗ ◦ + , µ + 1 2 q ∗⊤ ◦ B ∅ + min diag(σ ∅ ) B ⊤ ∅ q ∗ ◦ + , µ (5.21a) + 1 2 q ∗⊤ ◦ B ⊖ + min diag(σ ⊖ ) B ⊤ ⊖ q ∗ ◦ + , µ . (5.21b) Observethat(5.21b)isstronglyzeroduetothe(·) + operation. Further,(5.21a)vanishesbecause B ⊤ ∅ q ∗ ◦ =0. Using(B ⊤ ⊕ q ∗ ◦ ) + =B ⊤ ⊕ q ∗ ◦ ,wethenobtain ⃗ E D (q ∗ ◦ )= 1 2 q ∗⊤ ◦ B ⊕ min diag(σ ⊕ )B ⊤ ⊕ q ∗ ◦ , µ − q ∗⊤ ◦ a ◦ . (5.22) Letusdefinesuchavector σ ′ thatsatisfies diag(σ ′ ) B ◦ ⊤ q ∗ ◦ + =min{diag(σ ) B ◦ ⊤ q ∗ ◦ + , µ }. Letusdefinesuchavector σ ′ ⊕ thatsatisfies diag(σ ′ ⊕ )B ⊤ ⊕ q ∗ ◦ =min{diag(σ ⊕ )B ⊤ ⊕ q ∗ ◦ , µ }. (5.23) 5.3NonlinearConduction 131 Observethatσ ′ ⊕ 4σ ⊕ duetothemin{·} operationandσ ′ ⊕ ≻ 0becauseB ⊤ ⊕ q ∗ ◦ ≻ 0andq ∗ ◦ is entrywisefiniteasitminimizes ⃗ E D (q ◦ ). Substituting(5.23)into(5.22),weobtain ⃗ E D (q ∗ ◦ )= 1 2 q ∗⊤ ◦ B ⊕ diag(σ ′ ⊕ )B ⊤ ⊕ q ∗ ◦ − q ∗⊤ ◦ a ◦ . (5.24) Sincea ◦ isfeasible,foreverynonzerocurrentsourcethereexistsatleastonedirectedpathto ground. Thus, under any configuration of currents that can keep voltages at the nodes finite, the edgeswithpositivevoltage-differencebuildaconnectedgraphwithground. Thus,knowingthat q ∗ ◦ is entrywise finite as it minimizes ⃗ E D (q ◦ ), underq ∗ ◦ voltage distribution, the set of edges withpositivecomponentinB ⊤ ⊕ q ∗ ◦ buildaconnectedgraphwithground,implyingthepositive definitenessof B ⊕ diag(σ ′ ⊕ )B ⊤ ⊕ . Thisimpliesthatthefunctional 1 2 q ◦ ⊤ B ⊕ diag(σ ′ ⊕ )B ⊤ ⊕ q ◦ − q ◦ ⊤ a ◦ (5.25) is strictly convex in q ◦ and so finds its minimum at the critical point, where its first order variationvanishes. Comparing(5.25)with (5.24),theminimizingq ∗ ◦ mustsatisfy a ◦ =B ⊕ diag(σ ′ ⊕ )B ⊤ ⊕ q ∗ ◦ . (5.26) In light of (5.23) and by utilizing the identities (B ⊤ ⊖ q ∗ ◦ ) + = (B ⊤ ∅ q ∗ ◦ ) + = 0 and B ⊤ ⊕ q ∗ ◦ = (B ⊤ ⊕ q ∗ ◦ ) + ,onecanrephraseequation (5.26)as a ◦ =B ◦ min diag(σ ) B ◦ ⊤ q ∗ ◦ + , µ = ⃗ L ◦ (q ∗ ◦ ) which recovers the Poisson equation (5.18) atq ∗ ◦ . Further,q ∗ ◦ is unique as itneeds to minimize thestrictlyconvexfunctional (5.25). At first glance the Dirichlet principle reveals very little about minimizing dissipative energy onaresistivenetwork. Theconnection,however,ismadebythefactthatThomson’sprinciple is indeed the dual of Dirichlet’s principle. The next theorem extends this fact to capacitated directed graphs by defining nonlinear Thomson principle and showing that it is the dual of nonlinearDirichletprinciplewithzerodualitygap. Theorem 5.2. (Nonlinear Thomson Principle) Consider a nonlinear resistive network with capacitated directed edges. Under a vector of feasible current sources a ◦ , minimizing the 132 MulticlassMinimumCostRouting nonlinear Dirichletenergy (5.19)equallyleadstononlinearThomsonprinciple,expressedby min ⃗ E R (f):=f ⊤ diag(σ ) − 1 f s.t. 04f4µ , B ◦ f =a ◦ (5.27) whichreadsthetotalenergydissipationminimizationonthenetworkundertheconstraintsof directionalityandcapacityontheedgesandKCLatthenodes. Further,voltagesatthenodes actasthe Lagrangemultipliersforthedualoftheoptimizationproblem (5.27). Proof. Considering(5.27)astheprimaloptimizationproblem,obtainitsdualproblemas max λ min 04f4µ L(λ ,f):=f ⊤ diag(σ ) − 1 f +2λ ⊤ o a ◦ − B ◦ f whereλ is the vector of Lagrange multipliers. The first order condition ∇ f L =0 results in f opt =diag(σ )B ◦ ⊤ λ ,whereenforcingtheconstraint04f4µ leadsto f opt =min diag(σ ) B ◦ ⊤ λ + , µ . (5.28) MultiplyingbothsidesbyB ◦ andutilizingtheconstraintB ◦ f =a ◦ ,weobtain a ◦ =B ◦ min diag(σ ) B ◦ ⊤ λ + , µ = ⃗ L ◦ (λ ) which recovers the nonlinear Poisson equation (5.18). By Th. 5.1, on the other hand, (5.18) has auniquesolutionthatminimizesthenonlinearDirichletenergy (5.19). Thus,λ represents the uniquevectorofnodevoltagesthatminimize (5.19). Suchvoltagesinduceedgecurrentsthat followthenonlinearOhmlaw(5.17),viz., f =min diag(σ ) B ◦ ⊤ λ + , µ . (5.29) Comparing (5.28) against (5.29) implies that the edge currentsf opt that solve the nonlinear Thomsonprinciple (5.27)leadtosuchnodevoltagesλ thatminimizethenonlinearDirichlet energy (5.19),whichalsoplaythenaturalroleoftheLagrangemultipliers. Remark 5.1. Ondirectedbutuncapacitatedgraphs,whereµ ij =∞foralledges,ourresults inthissectionreducetothoseofCh.3. Specifically,thenonlinearDirichletLaplacian ⃗ L ◦ (q ◦ ) in (5.18) simplifies to ⃗ L ◦ q ◦ in (3.47), the nonlinear Dirichlet principle of Th. 5.1 reduces to thatofTh.3.6,andthenonlinearThomsonprincipleofTh.5.2reducestothatofTh.3.7. 5.4Multi-chargeNonlinearConduction 133 5.4 Multi-chargeNonlinear Conduction In a classical resistive network, the total charge generated by all current sources is absorbed by onesinglegroundednodeasthesink. Amorecomplexscenario,however,maybeenvisioned inparallelwithmulticlassproblemsindatanetworking. Weconsideramulti-chargeresistive network in which different typesof electrical charges are generated by the current sources and each typeofcharge canbeabsorbedbyaspecificnodeasthesinkforthatcharge. Letusrefer totheelectricalchargeabsorbablebyanoded asd-charge. Withoutedgecapacityconstraints,eachtypeofchargehasitsownindependentelectrical conduction,sothatthemulti-chargeresistivenetworkcanbeviewedasthecollectionoffully decoupled uni-charge resistive networks with no mutual interaction. When the edges are of limitedcapacities,however,theconductionofdifferentchargesnolongerhappensindependently, as the way of allocating edge capacities to each charge has a direct impact on the electrical conductionofthatcharge,whilethe sumof allocatedcapacities oneach edge isbounded. For example, allocating the total capacity of one edge to only one charge means eliminating that edgefromthenetworkforallother charges. Consider a multi-charge resistive network(V,E,K) constrained by both edge directionality and capacity. Letus introduce anedge capacityfactor06θ (d) ij 61 that determinesthe portion of capacity of each edge devoted to each charge. Thus, for a charged, available capacity on an edgeij isdeterminedby µ (d) ij =θ (d) ij µ ij with P d∈K θ (d) ij 61, ∀ij∈E, ∀d∈K. (5.30) Let us composeθ (d) ∈R |E| as the vector of edge capacity factors for the charge d. We also endow edges with the possibility of having different conductivities for different charges and composeσ (d) ∈R |E| as the vector ofd-conductivities on the edges. If one can determine the vectorθ (d) foreachcharged,thentheelectricalconductionofeachchargecomplieswiththe nonlinearuni-chargeequations(5.17)–(5.18). Letuscomposehyper-vectors o q ◦ asmulti-chargevoltagesatthenodes,f f asmulti-charge currentsthroughtheedges,and o a ◦ asmulti-chargeindependentcurrentsourcesinjectedinto thenodes, all conformablystructuredastheircounterpartsdefinedinSec.5.2.4. Likewise,let uscomposethehyper-vectorsofedgecapacityfactorsandedgeconductivitiesas θθ :=vec θ (1) ,...,θ (|K|) ∈R |E||K| o σ :=vec σ (1) ,...,σ (|K|) ∈R |E||K| . 134 MulticlassMinimumCostRouting Thenthemulti-chargeelectricalcurrentsthroughthecapacity-constraineddirectededgesare determinedbythefollowingmulticlassnonlinearOhmlaw: f f =min diag(o σ ) IB ◦ ⊤ o q ◦ + , θθ ⊙ (1 |K| ⊗ µ ) . (5.31) The term (1 |K| ⊗ µ ) expandsµ ∈ R |E| to be of size|E||K| for being used in a multi-charge fashion, where its entrywise product with θθ shapes (5.30) in a hyper-vector form. Plugging the multi-class nonlinear Ohm law (5.31) into Kirchhoff’s current law (5.12), we obtain the followingmulticlassnonlinearPoissonequationoncapacitateddirectednetworks: − I ⃗ L ◦ (o q ◦ , θθ )+ o a ◦ =0 I ⃗ L ◦ (o q ◦ , θθ ):=IB ◦ min diag(o σ ) IB ◦ ⊤ o q ◦ + , θθ ⊙ (1 |K| ⊗ µ ) . (5.32) WerefertoI ⃗ L ◦ (·)asmulticlassnonlinearDirichletLaplacianwhichactsonacapacitated directedgraphwith|K| differentclasses. Onecanverifythat I ⃗ L ◦ (o q ◦ , θθ )isnothingmorethan theconcatenationofuniclassnonlinearDirichletLaplaciansofdifferentclasses: I ⃗ L ◦ (o q ◦ , θθ )=diag ⃗ L (1) ◦ (q (1) ◦ ,θ (1) ),..., ⃗ L (|K|) ◦ (q (|K|) ◦ ,θ (|K|) ) ∈R (|V|− 1)|K|× (|V|− 1)|K| ⃗ L (d) ◦ (q (d) ◦ ,θ (d) )=B (d) ◦ min diag(σ (d) ) B (d)⊤ ◦ q (d) ◦ + ,θ (d) ⊙ µ , ∀d∈K. Observe that for each classd, the nonlinear Dirichlet Laplacian ⃗ L (d) ◦ (·) follows the definition of ⃗ L ◦ (·) in (5.18)foradirectednetworkwiththeedgecapacitiesofθ (d) ⊙ µ . Similar to the uni-chargecase, for the multi-charge Poisson equation (5.32) to have afinite solution,twoconditionsneedtobemet: First,everynonzerocurrentsourceofeachtypemust connect to its corresponding sink through at least one directed path. Second, there must exist at least one configuration of multi-charge edge currents that can transfer currents from all sources totheircorrespondingsinkswhilethesumofcurrentsofmultiplechargesoneachedgedoes notexceedthecapacityofthatedge. Intheabsenceofeitherofthesetwobasicconditions, on theotherhand,thenetworkflowproblemhasindeednoadmissiblesolution. Definition 5.2. Avectorofmulti-chargecurrentsourcesisfeasibleifthereexistsatleastone configurationofmulti-chargeedgecurrentssuchthatKCLissatisfiedforeachchargeateach nodeswhilethecumulativecurrentof multiplechargesrespectthecapacityofeachedge. Weknowthatthe|K|conductionprocesses (5.31)–(5.32)arenotindependent,buttightly coupled together due to the sharing of limited edge capacities. Then the crucial question is how 5.4Multi-chargeNonlinearConduction 135 to optimally allocate edge capacities to different charges. To answer this question, we begin withDirichlet’sprincipleonamulti-chargenonlinearresistivenetwork. Theorem 5.3. (MulticlassNonlinearDirichletPrinciple) Consider a multi-charge nonlinear resistive network with capacitated directed edges. With fixed edge capacity factors θθ and under a feasible multi-charge current sources o a ◦ , the multiclass nonlinear Poisson equation (5.32) has auniquesolutionthatminimizesthemulticlassnonlinearDirichletenergy ⃗ E D (o q ◦ ):= 1 2 o q ◦ ⊤ I ⃗ L ◦ (o q ◦ , θθ )− o q ◦ ⊤ o a ◦ . (5.33) Proof. With a fixed θθ , the sub-networks related to different charges d ∈ K behave as fully independent systems. Thus, to minimize (5.33), it suffices to separately minimize for each charged∈K,thecorrespondinguniclassnonlinearDirichletenergy ⃗ E (d) D (q (d) ◦ )= 1 2 q (d)⊤ ◦ ⃗ L (d) ◦ (q (d) ◦ ,θ (d) )− q (d)⊤ ◦ a (d) ◦ . (5.34) Further,Th.5.1ensuresthattheuniclassnonlinearPoissonequation − ⃗ L (d) ◦ (q (d) ◦ ,θ (d) )+a (d) ◦ =0 (5.35) has a unique solution that minimizes (5.34). Then the proof is concluded by observing that (5.32)isnothingexcepttheconcatenationof (5.35)fordifferentclasses. ThecrucialpointaboutTh.5.3isthefactthatthesolutionofPoissonequation (5.32)and the minimum possible value of multiclass Dirichlet energy (5.34) are both functions of edge capacityfactors θθ . Amongdifferentpossible θθ ’s,weareinterestedinthosethatpush ⃗ E D (o q ◦ ) toitsminimumunderafeasible o a ◦ . Thus,theoptimal θθ isdefinedby θθ ∗ :=argmin 1 2 o q ◦ ⊤ I ⃗ L ◦ (o q ◦ , θθ )− o q ◦ ⊤ o a ◦ s.t. 1) − I ⃗ L ◦ (o q ◦ , θθ )+ o a ◦ =0 2) P d∈K θ (d) 41 |E| , θθ <0. (5.36) It is important to understand that while the optimal o q ◦ which minimizes ⃗ E D (o q ◦ ) is unique, the related θθ is not necessarily unique, i.e., different θθ may lead to the same optimal o q ◦ . In Fig 5.2, for example, assume that nodes injects two types of electrical charge with intensity 3, one destined for node a and another for node b. Let all edges be of unit conductivity and 136 MulticlassMinimumCostRouting a c 2 2 1 1 1 1 3 3 s b Fig.5.2 Nodes injects two types of electrical charge with intensity 3, one destined for nodea and anotherfornodeb. Allnonlinearresistorsareofunitconductanceandofcapacity5. Electrical currentsthrough the edges follow the multi-chargenonlinearOhmlaw(5.31). ofcapacity5. Itiseasytoconfirmthatforminimizing ⃗ E D (o q ◦ ),twounitsofa-current(resp., b-current) should be sentthrough edgesa (resp., edgesb) and one unit through edgessc and ca(resp.,edgesscandcb). Thus,anydivisionofedgecapacitiesbetweenthetwochargescan solve (5.36) as far as it can provide for the chargea (resp., thechargeb) thecapacity of atleast 2onedgesa(resp.,edgesb)andthecapacityofatleast1onedgesscandca(resp.,edgessc andcb). For instance, any of the following combinations can be chosen for the capacity factors ofedgesa:{θ (a) sa =2 andθ (b) sa =3},{θ (a) sa =5 andθ (b) sa =0}or{θ (a) sa =3 andθ (b) sa =1}. The upshot of this section is formalized by the next theorem that extends the notion of Thomson’sprinciple, anditsimportantimplicationofminimizingquadraticdissipativeenergy, tomulti-chargeconductiononcapacitateddirectedgraphs. Wedrawthereader’sattentionin advance that despite Th. 5.3 which discusses the minimization of nonlinear Dirichlet energy under fixed edge capacity factors, the next theorem claims the global minimization of total dissipative energy with respect to both edge capacity factors and edge current distributions, whicharealsorelatedtothenodevoltagedistributionsbythenonlinearOhmlaw(5.31). Theorem5.4. (MulticlassNonlinearThomsonPrinciple)Consideramulti-chargenonlinear resistive network with capacitated directed edges. Under a feasible multi-charge current sources a ◦ , minimizing the multiclass nonlinear Dirichlet energy (5.33) equally leads to multiclass nonlinear Thomsonprinciple,expressedby min ⃗ E R (f f):=f f ⊤ diag(o σ ) − 1 f f s.t. 04 P d∈K f (d) 4µ , IB ◦ f f = o a ◦ (5.37) whichreadsthetotalenergydissipationminimizationonthenetworkundertheconstraintsof directionalityandcapacityontheedgesandKCLatthenodesforeachcharge. Further,voltages atthenodesactastheLagrangemultipliersforthedualoftheoptimizationproblem (5.37). 5.5Dirichlet-BasedRoutingPolicy 137 Proof. Letf f ∗ betheoptimalsolutionfor (5.37),inducingthemulti-chargevoltages o q ∗ ◦ atthe nodes. Thus,theedgecurrentsf ∗ (d) mustminimizethetotalenergydissipation ⃗ E (d) R (f (d) ) for each charged. Then Th. 5.2 implies that the node voltagesq ∗ (d) ◦ act as the Lagrange multipliers forthedualof ⃗ E (d) R (f (d) ) minimizationproblemandcanbefoundfrom q ∗ (d) ◦ =argmin q (d) ◦ 1 2 q (d)⊤ ◦ ⃗ L (d) ◦ (q (d) ◦ )− q (d)⊤ ◦ a (d) ◦ , ∀d∈K. Concatenatingthelatterforallchargesd∈K impliesthatthemulti-chargenodevoltages o q ∗ ◦ actastheLagrangemultipliersforthedualofoptimizationproblem (5.37)andminimizethe multiclassnonlinearDirichletenergy ⃗ E D (o q ◦ ). 5.5 Dirichlet-BasedRouting Policy On multiclass data networks, BP-based schemes transmit packets from only one class over each activated link at each timeslot, with two negative consequences: First, when the number of packetsfromindividualclassesisnotenoughtofillupthechannelcapacities,networkresources aresquandered. Inotherwords,more network capacitycould beutilized, andso theaverage networkdelaycoulddecrease,ifthecapacityofeachchannelwereproperlystuffedwithpackets from different classes. Second, even if an individual class with the largest queue differential has enoughpacketstofillupalink,blindlystuffingthelinkwithoneclasscouldonlydepletethe network resourceswithevennegativeimpactondelayperformance[103]. We propose a multiclass routing protocol in an attempt to answer the question of how a dynamicroutingpolicy,withnopredefinedroutingpath,canmaximizetheutilizationofnetwork resourcesateachtimeslot. Oursolutionisinspiredbywhatintroducedintheprevioussection as the multi-charge electrical conduction on nonlinear resistive networks, which hints us on three things: First, a dissipative energy minimizing policy cannot be limited to sending packets from only one class over each activated link. Second, the optimal capacity allocation (5.36) suggests that the piece of channel capacity devoted to each class should be properly sized by its linkqueue-differentialandprofit-factor,consideredasreciprocalofthelinkcostfactor. Third, the nonlinear Ohm laws (5.17) and (5.31) suggest that rather than transmitting as many packets aspossibleovereachactivatedlink,thenumberofforwardedpacketsshouldlimittothelink queue-differentialscaledbythelinkprofit-factor. 138 MulticlassMinimumCostRouting On multiclass networks, at each timeslot n, the Dirichlet-based routing policy observes queue backlogsq (d) i (n) andestimateschannelcapacitiesµ ij (n) andlinkcostfactorsρ (d) ij (n) for eachclasstomakeanetwork-layerroutingdecisionasfollows. 1) Multiclass Dirichlet-based weighting: On every directed linkij and for each classd find q (d) ij (n):=q (d) i (n)− q (d) j (n)anddetermine ̂ f (d) ij (n)bysolvingthefollowinglocalconstrained optimizationproblem: ̂ f (d) ij (n):=argmin X d∈K q (d) ij (n) ρ (d) ij (n)− ̂ f (d) ij (n) 2 s.t. 1) P d∈K ̂ f (d) ij (n)6µ ij (n) 2) 06 ̂ f (d) ij (n)6q (d) i (n), ∀d∈K (5.38) Thengiveaweighttoeachclassd∈K ij (n) as w (d) ij (n):=2q (d) ij (n) ̂ f (d) ij (n) ρ (d) ij (n)− ̂ f (d) ij (n) 2 (5.39) andaggregatethemtodeterminethefinallinkweightas w ij (n):= X d∈K w (d) ij (n). (5.40) 2) MulticlassDirichlet-basedscheduling: Findtheschedulingvector,inthesamewayasBP, usingthemax-weightscheduling (5.8). 3) Multiclass Dirichlet-based forwarding: Transmit ̂ f (d) ij (n) number of packets from the classd overeachactivatedlinkij. We refer to ̂ f (d) ij (n) as link transmission prediction. It represents the number ofd-packets that link ij would transmit if it were activated at slot n, which is a predicted value that will berealizedonlyiflinkij isselectedintheschedulingstep. Tounderstandthealgorithm,itis important to discriminate link transmission prediction from the link actual transmissionf (d) ij (n), which is the realization of link transmission prediction if the link is scheduled, and the channel capacityµ ij (n),whichisthemaximumnumberofpacketsthelinkcantransmit. Table 5.1 compares the architecture of multiclass V-parameter BP with our proposed Dirichlet-basedpolicy. Theyenjoythesamealgorithmicstructure,computationalcomplexity andoverheadsignaling,whichprovidesa convenientwayofunifyingthe newpolicywiththe extensiveworkspreviouslydonearoundBP. 5.5Dirichlet-BasedRoutingPolicy 139 Table 5.1 Comparing algorithmicstructureofmulticlassV-parameterBPandDirichlet-based routing policies, emphasizing thesamecomputationalcomplexityandoverheadsignaling. Weighting V-BP d ∗ ij (n)=argmax d∈K q (d) ij (n) ̂ f (d) ij (n)= ( min{µ ij (n), q (d) i (n)} if d=d ∗ ij (n) 0 otherwise w ij (n)=µ ij (n) q (d ∗ ) ij (n)− Vρ (d) ij (n)µ ij (n) + Dirichlet ̂ f (d) ij (n)=argmin X d∈K q (d) ij (n) ρ (d) ij (n)− ̂ f (d) ij (n) 2 s.t. P d ̂ f (d) ij (n)6µ ij (n) & 06 ̂ f (d) ij (n)6q (d) i (n) w ij (n)=2q (d) ij (n) ̂ f (d) ij (n) ρ (d) ij (n)− ̂ f (d) ij (n) 2 Scheduling π (n)=argmax π ∈Π X ij∈E π ij w ij (n) Forwarding f (d) ij (n)=    ̂ f (d) ij (n) if π ij (n)=1 0 otherwise Remark 5.2. Onuniclassnetworks,linktransmissionpredictionsandlinkweightsreduce to ̂ f ij (n)=min q ij (n) + ρ ij (n), µ ij (n) w ij (n)=2q ij (n) ̂ f ij (n) ρ ij (n)− ̂ f ij (n) 2 whichrecoversthePareto-optimalHDpolicywithβ =1,proposedinCh.3. Remark 5.3. LikeBP,ouralgorithmalsorestsonacentralizedschedulingwhosecomplexity can be prohibitive in practice. Fortunately, much progress has been made to ease this difficulty bydesigningdecentralizedschedulerswithanarbitrarytradeoffbetweencomplexityandvicinity tothecentralizedperformance(seeGupta07,Bui09,Jiang11andreferencestherein). Optimization problem (5.38), which determines the link transmission predictions, is a least- normproblemwithvariableboundsthatcanbesolvedinfastpolynomialtimeateachindividual node,i.e.,inafullydecentralizedmanner. Hereweproposearelatedalgorithmthatdoes so. Tosimplify notation, wedroptimeslotvariable(n). Firstobservethatinsolving(5.38),itis trivial thatone needsto take ̂ f (d) ij =0 for eachclassd withq (d) ij 60. Thus, let uscreate the set K ij ⊆K suchthatq (d) ij > 0,∀d∈K ij andfix ̂ f (d) ij =0,∀d / ∈K ij . Letustemporarilyneglect 140 MulticlassMinimumCostRouting ෢ ( 1 ) + ෢ ( 2 ) = = ൗ ( 1 ) ( 1 ) , ൗ ( 2 ) ( 2 ) = 6 ෢ ( 2 ) ෢ ( 1 ) Unique optimal solution without integer constraints One optimal solution under integer constraints ෢ ( 1 ) + ෢ ( 2 ) = = ൗ ( 1 ) ( 1 ) , ൗ ( 2 ) ( 2 ) = 6 ෢ ( 2 ) ෢ ( 1 ) Unique optimal solution without integer constraints One optimal solution under integer constraints ෢ ( 1 ) + ෢ ( 2 ) = = ൗ ( 1 ) ( 1 ) , ൗ ( 2 ) ( 2 ) = 6 ෢ ( 1 ) Unique optimal solution without integer constraints One optimal solution under integer constraints Unique optimal solution without integer constraints Two optimal solutions under integer constraints ෢ ( 1 ) + ෢ ( 2 ) = = ൗ ( 1 ) ( 1 ) , ൗ ( 2 ) ( 2 ) = 6 ෢ ( 1 ) ෢ ( 2 ) ෢ ( 2 ) Fig.5.3Geometricalinterpretationofsolvingoptimizationproblem (5.38)foratwo-classcase. The figures on top are examples with the channel capacity excess e< 0 and on bottom withe> 0. The top-right example illustrates that the optimal integer solution is not necessarily unique, as there could exist more than one vertex of the unit hypercube (square here) with equal shortest distancefrom the initial unique non-integersolution. thatthesolutionmustbeintegerand definethechannelcapacityexcess e:=µ ij − X d∈K ij q (d) ij ρ (d) ij . (5.41) Thepivotisthesignofe. Fore>0, problem (5.38)findsitsminimumofzeroatpoint p:= q (1) ij ρ (1) ij ,··· ,q (|K ij |) ij ρ (|K ij |) ij . (5.42) Otherwise,withe<0,thefirstconstraintin (5.38)changesfrominequalitytoequality. Then a basic Lagrange argument shows that in the absence of lower variable bounds and integer 5.5Dirichlet-BasedRoutingPolicy 141 constraints,theproblemhasauniquesolutionas ̂ f (d) ij =q (d) ij ρ (d) ij +e |K ij |, ∀d∈K ij . (5.43) Further,ifforaclassd ′ ∈K ij weget ̂ f (d ′ ) ij 60,thezerolowerboundentailstoeliminated ′ from K ij andrecalculate (5.43)basedonnewe. Notethatifsomed ′ classesareremovedfromK ij , still the newe remains negative. To see this, letK ′ ij be the set of classes removed fromK ij and letK ′′ ij :=K ij \K ′ ij betheremainingsetofclasses. Then (5.41)leadsto X d∈K ′′ ij q (d) ij ρ (d) ij =µ ij − e− X d ′ ∈K ′ ij q (d ′ ) ij ρ (d ′ ) ij . Since ̂ f (d ′ ) ij 60 entailsq (d ′ ) ij ρ (d ′ ) ij 6− e |K ij |,weobtain X d∈K ′′ ij q (d) ij ρ (d) ij >µ ij − e+e|K ′ ij | |K ij |>µ ij wherethelastinequalityistheresultofe<0. Bringing integer constraints to the picture, let us first consider the case of negative e. By geometry,(5.43)readstheprojectionofpointp ontothehypersurface X d∈K ij ̂ f (d) ij =µ ij . (5.44) Underintegerconstraints,thehyperspaceisreplacedbyanintegerlatticeofdimension|K ij |. Optimal solution to the integer problem is then an element of this lattice that has the minimum Euclidean distance from pointp. Further, such an element must be one of the vertices of a unit hypercube with integer coordinates that contains the non-integer solution (5.43). Thus, among verticesofthishypercubethatlieonthelattice,theoptimalintegersolutionistheoneclosest to the projection of pointp onto the hypersurface (5.44). A graphical demonstration of this conceptisillustratedinFig.5.3foranexampleoftwoclassesinK ij . Toformalizethisgeometricalprocedure,letusdefinethefollowinglinkresiduals r (d) := ̂ f (d) ij − ̂ f (d) ij , ∀d∈K ij r := X d∈K ij r (d) =µ ij − X d∈K ij ̂ f (d) ij (5.45) 142 MulticlassMinimumCostRouting where the last equality results from (5.44). Then the above geometry-based approach of finding optimalintegersolutionfore<0 leadsto ̂ f (d) ij =    ̂ f (d) ij forr classesinK ij withlargestr (d) ̂ f (d) ij fortherestofclassesinK ij . (5.46) Let us add integer constraints to the simpler case of e > 0. Again, the integer solution must be one of the vertices of a unit hypercube with integer coordinates that contains the initial solution (5.42). If this hypercube does not intersect the hypersurface (5.44), the optimal integer solution is simply the vertex with the minimum Euclidean distance from pointp. If the intersection exists, however, the solution must be sought among those vertices that meet thechannelcapacityconstraint. Translatingthistoalgebra,ifthechannelcapacityallows,the optimalintegersolutionisobtainedbyroundingthenon-integersolution (5.42)tothenearest integer. If the resulted integer exceeds the channel capacity, on the other hand, one needs to rounddown,ratherthanroundup,enoughcomponentsofthenon-integersolutionuntilmeeting thechannelcapacityconstraint. Usingthedefinitionoflinkresidualsin (5.45)andtaking ̂ f (d) ij equaltothecomponentsofp in (5.42),wegettheoptimalintegersolutionfore>0 as ̂ f (d) ij =    ̂ f (d) ij foruptor classesinK ij withlargestr (d) >0.5 ̂ f (d) ij fortherestofclassesinK ij . (5.47) The complete procedure of determining transmission predictions for each classd∈K on a linkij issummarizedbyAlg.5.1. Remark5.4. Whenallwirelesschannelstakeunitcostfactorsforallclasses,theDirichlet-based policyofSec.5.5simplifiestotheminimum-delaypolicywithMCLTinCh.4,whichminimizes the average network delay in a class of routing policies that act based only on current queue congestionandcurrentchannelstates. 5.6 RoutingCostAnalysis In Sec. 5.3, wedeveloped a multi-charge electrical conductionon nonlinear resistive networks thatimposebothdirectionalityandcapacityconstraintson branches. InSec. 5.5,ontheother hand,wedevelopedanetworklayerroutingpolicyformulticlassmultihopwirelessnetworks influencedbyrandomarrivals,inter-channelinterferenceandtime-varyingtopology. Theformer 5.6RoutingCostAnalysis 143 Algorithm5.1Solvingproblem (5.38)foralinkij atslotn. CreateasetK ij (n)⊆K suchthatq (d) ij (n)>0,∀d∈K ij (n) Assign ̂ f (d) ij (n)=0,∀d / ∈K ij (n) if e:=µ ij (n)− P d∈K ij (n) q (d) ij (n)<0 S1: Set ̂ f (d) ij (n)=q (d) ij (n) ρ (d) ij (n)+e |K ij (n)|,∀d∈K ij (n) S2: Foranyd ′ ∈K ij (n) with ̂ f (d ′ ) ij (n)<0,assign ̂ f (d ′ ) ij (n)=0, removed ′ fromK ij (n),andgobacktoS1 S3: Letr (d) := ̂ f (d) ij (n)− ̂ f (d) ij (n) ,r:= P d r (d) ,∀d∈K ij (n) S4: Assign ̂ f (d) ij (n),∀d∈K ij (n),accordingto(5.46) else S5: Set ̂ f (d) ij (n)=q (d) ij (n) ρ (d) ij (n),∀d∈K ij (n) S6: Letr (d) := ̂ f (d) ij (n)− ̂ f (d) ij (n) ,r:= P d r (d) ,∀d∈K ij (n) S7: Assign ̂ f (d) ij (n),∀d∈K ij (n),accordingto(5.47) end describes a deterministic continuous-time process, while the latter leads to a stochastic slotted- time process. This section explains how these two seemingly different problems are indeed rigorouslycorrelatedwitheachother. Weparticularlyshowthatinalong-termaveragebasis, multiclasspacketflowonawirelessnetworkgovernedbyourroutingprotocolcomplieswith multi-chargeconductioncurrentonasuitably-definednonlinearresistivenetwork,wherethe pivotisthenotionoffluidlimitofastochasticprocess. 5.6.1 Dirichlet-basedtimeslotbehavior Givennetworkconditionandqueuecongestion,everyroutingpolicyhasitsownspecificwayof forwarding packets overthe network. This defines timeslot characteristic of a routing policy, which in turn shapes traffic behavior on a network controlled by that policy. We formalize such acharacteristicforourDirichlet-basedroutingpolicyinthenexttheorem,whichiskeytothe proof of Th. 5.6 on routing cost minimization and Th. 5.7 on network throughput optimality. 144 MulticlassMinimumCostRouting Priortothat,letusdefinethehyper-vectoroflinkcostfactorsas o ρ (n):=vec ρ (1) (n),...,ρ (|K|) (n) ∈R |E||K| ρ (d) (n):= ρ (d) 1 (n),...,ρ (d) |E| (n) ⊤ . (5.48) Theorem 5.5. (Timeslot Behavior) At each timeslot n, the Dirichlet-based routing policy minimizes thef f-controlled functional∥diag(o ρ (n)) − 1 IB ◦ ⊤ o q ◦ (n)− f f(n)∥ subject to network constraints,includingdirectionality,capacityandinterference. Proof. We need to show that the link forwarding by the Dirichlet-based routing algorithm satisfiesthefollowingintegeroptimizationproblemateachslot n: min f (d) ij (n) X ij∈E X d∈K q (d) ij (n)/ρ (d) ij (n)− f (d) ij (n) 2 s.t. 1) f (d) ij (n)>0, ∀d∈K 2) P d∈K f (d) ij (n)6µ ij (n) 3) P b∈out(i) f (d) ib (n)6q (d) i (n), ∀d∈K 4) Interference Model (5.49) wherethefirstconstrainttreatslinkdirectionality,thesecondconstraintensureschannelcapacity, thethirdconstraintreadsthefactthatthenumberofd-packetsleavinganodecannotbelarger thanthequeuebacklogofclassdatthatnode,andthefourthconstraintimposestherequirement ofinterferencemodelonsimultaneousactivationofdifferentlinks Firstobservethatgivencurrentq (d) ij (n),thefollowingidentityistrue: argmin f (d) ij X d∈K q (d) ij /ρ (d) ij − f (d) ij 2 =argmax f (d) ij X d∈K 2q (d) ij f (d) ij /ρ (d) ij − f (d) ij 2 . Now assuming an interference model under which a node can transmit to only one neighbor at a timeslot, the third constraint in (5.49) reduces tof (d) ij (n)6q (d) i (n),∀d∈K. This way, one can decouple the impact of interference from other constraints. In other words, it allows to solvethe problem in two independent stepsof firstlocallyweightingeach link andthen globally schedulingasetoflinkswithminimumtotalweight. Thenthelocallinkweightingisformalized 5.6RoutingCostAnalysis 145 bythedistributedoptimizationproblem w opt ij (n):=max f (d) ij (n) X d∈K 2q (d) ij (n)f (d) ij (n)/ρ (d) ij (n)− f (d) ij (n) 2 s.t. 1) P d∈K f (d) ij (n)6µ ij (n) 2) 06f (d) ij (n)6q (d) i (n), ∀d∈K (5.50) thatfindstransmissionpredictionsforeachoutgoinglink ij connectedtothenodei. Further, theglobalschedulingisformalizedbythecentralized optimizationproblem π opt (n):=argmax π ∈Π X ij∈E π ij w opt ij (n). (5.51) The proof is concluded by observing that (5.51) reads the max-weight scheduling, and that w ij (n) derivedby (5.38)–(5.40)complieswithw opt ij (n) assignedby (5.50). Remark 5.5. On uniclass networks, Th. 5.5 reduces to Th. 3.1 for β = 1 in Ch. 3, stating that at each timeslot n, the Pareto optimal HD policy with β = 1 maximizes the functional 2f(n) ⊤ ρ (n)B ◦ ⊤ q ◦ (n)− f(n) ⊤ f(n) subjecttonetworkconstraints. Next assumption is being used in the analysis of Dirichlet-based routing properties, stating that the greedy minimization of∥diag(o ρ (n)) − 1 IB ◦ ⊤ o q ◦ (n)− f f(n)∥ at each timeslot leads to its minimum long-term average. Therefore, the Bellman’s principle of optimality, and so dynamic programming, can be used to minimize∥diag( o ρ ) − 1 IB ◦ ⊤ o q ◦ − f f∥. It also implicitly means that nooverlappingexistamongslot-basedsubstructuresofthelong-termminimizationproblem. Assumption5.1. Givenacombinationofnetworktopologyandmulticlasstrafficrates,timeslot minimization of functional∥diag(o ρ (n)) − 1 IB ◦ ⊤ o q ◦ (n)− f f(n)∥ subject to network constraints is anoptimalsubstructureforglobalminimizationofitslong-termaverage. 5.6.2 Dirichlet-basedfluidbehavior Fluidlimitofastochasticprocessisthelimitdynamicsobtainedbyscalingintimeandamplitude. Escapingallthedetails,weonlyprovideabriefexplanationofthetheorythatisrequiredfor ouranalysishereandrefertheinterestedreaderto[75,76]formorerigorousinsight. LetX(ω,t)bearealizationofacontinuous-timestochasticprocessX alonganarbitrary sample pathω, and define the scaled process X r (ω,t) :=X(ω,rt)/r for anyr >0. Then a deterministic function ˜ X(t) is called fluid limit if there exist a sequencer and a sample pathω 146 MulticlassMinimumCostRouting suchthat lim r→∞ X r (ω,t)→ ˜ X(t)uniformlyoncompactsets. Foranystableflownetwork, the existence of fluid limit is guaranteed if exogenous arrivals are of finite variance. It is further shownthateachfluidlimitisLipschitz-continuous,andsodifferentiable,almosteverywhere withrespecttoLebesguemeasureon[0,∞). It is proven that all fluid limits, which are scaled trajectories, reveal the same behavior when scaling factor lim r→∞ . Such a common behavior can be represented by some deterministic equations called fluid model. The practical importance of fluid model is the easy way that it provides to analyze the rate-level behavior of the original stochastic process. Thus, while fluid limit is the scaled process of the first-order continuous-time approximation for an arbitrary realization of the stochastic process, the fluid model is a (set of) fully deterministic, continuous- timeequation(s)thatmimicthesamebehavioroffluidlimits. Toestablishthefluidmodelofawirelessnetwork,firstnotethatthefluidtheoremisdefined for continuous-timestochasticprocesses, while thewireless network isa slotted-timeprocess. To resolve this issue, given a sample pathω, we extend a slotted-time process to be continuous- timeusingfirst-orderapproximationineachtimeslotinterval (n,n+1). Letcontinuous-time hyper-vectors o a tot ◦ (t) andf f tot (t) represent the cumulative node arrivals and link transmissions up to time t. Assuming normalized timeslots with the period of time unit and using linear interpolationineachtimeslot interval(n,n+1),theslotted-timedynamicsofwirelessnetwork aretransferredtotheapproximatecontinuous-timefluidequations o q ◦ (t)= o a tot ◦ (t)− IB ◦ f f tot (t) (5.52) o a tot ◦ (t)= o a ◦ t (5.53) f ˙ f tot (t)=f f(t) (5.54) with o a ◦ beingthetimeaverageexpectationof o a ◦ (n) asdefinedin (5.2). Note that the above fluid equations hold for any stable wireless network operating under an arbitrarynon-idlingroutingpolicy,whileeveryroutingpolicyhasitsownwayofdetermining f f(t). TomakethefluidmodelspecifictotheDirichlet-basedroutingpolicy,oneneedtoadd routing criteria (5.38)–(5.40) into the picture, which are already abstracted by Th. 5.5 that the algorithm minimizes∥diag(o ρ (n)) − 1 IB ◦ ⊤ o q ◦ (n)− f f(n)∥ at each slotn subject to network constraints. Transferring this to continuous-time domain using linear interpolation, we then 5.6RoutingCostAnalysis 147 obtainparticularfluidequationfortheDirichlet-basedroutingpolicyas f f(t)=argmin diag( o ρ ) − 1 IB ◦ ⊤ o q ◦ (t)− f f(t) s.t. NetworkConstraints (5.55) wheretheexistenceof o ρ isguaranteedby (5.4). Remark 5.6. The solution to(5.55) isuniqueby thereasonthatagiven o q ◦ (t) leadstoaunique IB ◦ ⊤ o q ◦ (t),andsotouniqueq (d) ij (t) components,asthematrixIB ◦ hasfullrowrank. We skippedmany detailsinderiving (5.52)–(5.55) for the sake of brevity and readability. The path, however, is very similar to that for uniclass networks that we built in Ch. 3. It is important though to understand that the long-term average effect of every constraint in wireless network and every action of routing policy—the Dirichlet-based here—is considered in the derivationofnetworkfluidmodel. Let us envision a multiclass wireless network with packets being routed under the Dirichlet- basedpolicy. Fromamicroscopicstandpoint,ateachtimeslot,thepolicydecidestoactivatea particularsetoflinkstoforwardaspecificnumberofpacketsfromeachclass. Notethatdue to the link interference, a link is not transmitting at every slot, but only at those slots that is chosentobeactivatedintheschedulingstep. Fluidtheory,ontheotherhand,ensuresthatthe limit flow on each link, roughly viewed as the total number of packets transmitted over the link duringalargeperiodoftimedividedbythetimeduration,complieswithf f(t)resultedfromthe fluid equations (5.52)–(5.55). Then the crucial question is “what this flow really looks like?” To answerthisquestion,wetakeareverseapproachbydeterminingwhatlim t→∞ f f(t)should looklikeforbeingametaphorofmulti-chargenonlinearconductionofSec.5.4. Foraflow f f,thekeypropertytomimicthemulti-chargenonlinearconductionisDirichlet’s principle. More precisely, lim t→∞ f f(t) should be the result of minimizing the multiclass nonlinearDirichletenergy ⃗ E D (o q ◦ ). Onamulti-chargenonlinearresistivenetworkwithfixed edge capacities, ⃗ E D (o q ◦ ) depends on edge capacityfactors θθ . Ona wireless network, however, channel capacities depend not only on channel state probabilities but also on scheduling policy. Toformalizethesedependencies,wefirstintroducethenotionofeffectivecapacity. Given an interference model, let T π (t) represent the cumulative time until t in which a schedulingvectorπ ∈Π hasbeenselectedbyaroutingpolicytostabilizeamulticlasstraffic matrix. Obviously, each routing policy leads toits own specific T π (t), which will determine the effectivelinkcapacityunderthatpolicy. 148 MulticlassMinimumCostRouting Definition 5.3. Under a sequence of wireless link scheduling, the effective capacity on each linkisthetimeaverageexpectationofcapacitymadegenuinelyavailableonthatlink,viz., µ eff :=limsup τ →∞ 1 τ X π ∈Π τ X n=0 T π (n)− T π (n− 1) π ⊙ E{µ (n)} whereµ eff denotesthevectorofeffectivelinkcapacities. LetU eff representthesetofallpossiblevectorsµ eff subjecttoagiveninterferencemodel, whichisfinite(butarbitrarilylarge)for afinitenetwork. Next,wedefine theoptimaleffective edgecapacitiesandedgecapacityfactorsinthesenseofDirichletas {µ opt eff , θθ opt , o q opt ◦ }:=argmin 1 2 o q ◦ ⊤ I ⃗ L ◦ (o q ◦ ,µ eff , θθ )− o q ◦ ⊤ o a ◦ s.t. 1) − I ⃗ L ◦ (o q ◦ ,µ eff , θθ )+ o a ◦ =0 2) P d∈K θ (d) 41 |E| , θθ <0 3) µ eff ∈U eff (5.56) whereI ⃗ L ◦ isthemulticlassnonlinearDirichletLaplacianwithdiag(o σ )=diag( o ρ ) − 1 , I ⃗ L ◦ (o q ◦ ,µ eff , θθ )=IB ◦ min diag( o ρ ) − 1 IB ◦ ⊤ o q ◦ + , θθ ⊙ (1 |K| ⊗ µ eff ) . Then the multiclass nonlinear Thomson principle of Th. 5.4 implies that the total energy dissipation ⃗ E R (f f)=f f ⊤ diag( o ρ )f f isminimizedbythemulticlassnonlinearOhmlaw f f opt =min diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ + , θθ opt ⊙ (1 |K| ⊗ µ opt eff ) . (5.57) Remark 5.7. Havingµ opt eff , θθ opt and o q opt ◦ determined by the optimization problem (5.56),f f opt minimizes∥diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f∥ 2 subjecttoedgedirectionalityandcapacity. Therefore,forfluidequations (5.52)–(5.54)tomimicthemulti-chargenonlinearconduction, f f(t) should converge tof f opt att→∞. The next theorem shows that our proposed routing protocolmakesthisindeedhappened. Inotherwords,weclaimthatundertheDirichlet-based routing policy, observing average packet flow of different classes in limit (macroscopic flow), it takes the form of multi-charge conduction current on the underlying capacitated directed graph. Remark 5.8. We emphasize that neither the aim is to solve the equations (5.56)–(5.57), nor our proposed routing protocol in Sec. 5.5 depends on such a solution. Instead, we seek to 5.6RoutingCostAnalysis 149 connecttheseequationstothelimitdynamicsofamulticlasswirelessnetworkcontrolledby the Dirichlet-based policy. One benefit of this connection, which shall partially be exploited in this paper, is to deduce the properties of our protocol with getting inspiration from classical properties of conduction current. The more important benefit, however, is to establish a new frameworkforanalyzingstochastic,slotted-time,multiclass,interferingnetworksingeneral. We use Lyapunov argument in the proof of routing cost minimization and throughput optimality,wherethefollowinglemmaplaysacrucialrole: Lemma 5.1. Givenaconnectedmulticlasswirelessnetwork,let IM ◦ (n):= IB ◦ IB ◦ ⊤ − 1 IB ◦ diag(o ρ (n)) − 1 IB ◦ ⊤ . (5.58) First, the hyper-matrixIM ◦ (n) is pseudo positive definite in the sense that all of its eigenvalues are positive andx ⊤ IM ◦ (n)x> 0 for any vectorx∈R (|V|− 1)|K| , with equality if and only if x=0. Second,foranyvectorx∈R (|V|− 1)|K| ,thefollowingidentityholds: IB ◦ ⊤ IM ◦ (n)x=diag(o ρ (n)) − 1 IB ◦ ⊤ x. (5.59) Third,thereexistssuchascalar16η 63 thatforanyvectorsx,y∈R (|V|− 1)|K| , x ⊤ IM ◦ (n) ⊤ +IM ◦ (n) y6η x ⊤ IM ◦ (n)y. (5.60) Proof. Observe thatIM ◦ (n)=diag M (1) ◦ (n),...,M (|K|) ◦ (n) with the block components M (d) ◦ (n):= B (d) ◦ B (d)⊤ ◦ − 1 B (d) ◦ diag ρ (d) (n) − 1 B (d)⊤ ◦ . ThentheproofisdirectlyconcludedfromLem.3.1,Lem.3.2andLem.3.3inCh.3thatshowed theclaimsaretrueforanyconnecteduniclasswirelessnetwork. Theorem5.6. (MulticlassMinimumCost)LetamulticlasswirelessnetworkmeetAssum.5.1 underastabilizabletrafficmatrix. ThentheDirichlet-basedroutingpolicysolvestherouting costminimizationproblem (5.1). Proof. Let o q ∗ ◦ (t) andf f ∗ (t) denote the solution of the fluid model (5.52)–(5.55) and let o q opt ◦ and f f opt denote the solution of the multi-charge conduction equations (5.56)–(5.57), which are constantandindependentoft. WeknowfromTh.5.4thatf f opt minimizesthetotalenergy 150 MulticlassMinimumCostRouting dissipation ⃗ E R (f f)=f f ⊤ diag( o ρ )f f . Observingthatthecostfunctioninproblem (5.1)reads lim t→∞ ⃗ E R f f ∗ (t) ,itsufficestoshowthat f f ∗ (t) asymptoticallyconvergestof f opt . ConsidertheLyapunovfunction Y(t):= o q ∗ ◦ (t)− o q opt ◦ ⊤ IM ◦ o q ∗ ◦ (t)− o q opt ◦ where IM ◦ = IB ◦ IB ◦ ⊤ − 1 IB ◦ diag( o ρ ) − 1 IB ◦ ⊤ ispseudopositivedefinitebyLem.5.1. Taking timederivativefromY(t) andutilizingproperty (5.60),thereexistssuch16η 63 that ˙ Y(t)6η o˙ q ∗ ◦ (t) ⊤ IM ◦ o q ∗ ◦ (t)− o q opt ◦ From (5.52)–(5.54) we get o˙ q ∗ ◦ (t)= o a ◦ − IB ◦ f f ∗ (t). From (5.31)–(5.32), on the other hand, we get o a ◦ =IB ◦ f f ∗ . Theseidentitiesalongwithproperty (5.59)leadto η − 1 ˙ Y(t)6 f f opt − f f ∗ (t) ⊤ diag( o ρ ) − 1 IB ◦ ⊤ o q ∗ ◦ (t)− o q opt ◦ . Onecanmultiplybothsidesbytwo,addandsubtracttheterms o q ∗ ◦ (t) ⊤ o q ∗ ◦ (t)+f f ∗ (t) ⊤ f f ∗ (t)and +o q opt⊤ ◦ o q opt ◦ +f f opt⊤ f f opt ontherightandrecasttheexpressiontoobtain 2η − 1 ˙ Y(t)6− diag( o ρ ) − 1 IB ◦ ⊤ o q ∗ ◦ (t)− f f opt 2 + diag( o ρ ) − 1 IB ◦ ⊤ o q ∗ ◦ (t)− f f ∗ (t) 2 (5.61a) − diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f ∗ (t) 2 + diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f opt 2 . (5.61b) Letuscharacterize (5.61a)onthewirelessnetworkgivenq ∗ ◦ (t). UnderAssum.5.1andin lightoftheparticularfluidequation (5.55)fortheDirichlet-basedroutingpolicy, theimmediate resultisthatforanyalternativeforwardingf f thatsatisfiesnetworkconstraints, diag( o ρ ) − 1 IB ◦ ⊤ o q ∗ ◦ (t)− f f ∗ (t) 2 6 diag( o ρ ) − 1 IB ◦ ⊤ o q ◦ (t)− f f 2 ateachtimet. Thef f opt obviouslymeetsthedirectionalityconstraintsduetothestructureof conductionequations(5.56)–(5.57). Hence,weobtain 2η − 1 ˙ Y(t)6− diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f ∗ (t) 2 + diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f opt 2 . (5.62) Next,letus characterize (5.62) intheframeworkof multi-chargeconductionequations(5.56)– (5.57). We know that given o q opt ◦ and under edge directionality and capacity constraints, the minimumof diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f 2 occursatf f =f f opt . Hence,weget(5.62)60,which 5.7ThroughputAnalysis 151 inlightof16η 63leadsto ˙ Y(t)60. ByLaSalle’sinvarianceprinciple,thelatterguarantees thatevery o q ∗ ◦ (t)asymptoticallyconvergestoaninvariantsetsubjectto ˙ Y(t)=0. Letusassume ˙ Y(t)=0. Then2η − 1 ˙ Y(t)6 (5.62)60 entails diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f ∗ (t) = diag( o ρ ) − 1 IB ◦ ⊤ o q opt ◦ − f f opt . Sincef f opt minimizestheright-handsideandisunique,weobtainf f ∗ (t)=f f ∗ =f f opt which concludes the proof. Note the intentionally dropped time variable(t) emphasizes thatf f ∗ (t) turnstobestationarybyconvergingtof f opt . Remark5.9. Toourknowledge,thisisthefirsttimethatanetwork-layerroutingprotocolasserts the strict minimization of a general routing penalty subject to network stability and without requiring any statistical information about network or traffic. Note that in V-parameter BP [ 63], the[O(V), O(1/V)] delay-cost tradeoffprevents minimizingcostsubject tostability,meaning thatnetworkdelaygrowstoinfinity asroutingcostispushedtowardsitsminimum. Remark 5.10. By Th. 5.6, minimum routing cost under the Pareto optimal HD policy of Ch. 3 is extended in two directions: First, Assum. 3.2 is removed and so is the main restriction of requiring the fluid limit to follow uncapacitated, rather than capacitated, diffusion equations. Second,theresultisdevelopedformulticlassnetworksthatalongsidethecapacityconstraints ondiffusionequationsmakethetheoryfarmorecomplicated. Remark 5.11. Considering Th. 5.6 and 5.4 together, one can extend the Pareto optimal results ofCh.3tomulticlasswirelessnetworksandwithoutrequiringAssum.3.2thatthefluidlimit hadtocomplywithuncapacitateddiffusionequations. 5.7 ThroughputAnalysis Throughputoptimality,asdefinedinSec.5.2.1,isanimportantqualitycharacteristicofDirichlet- basedroutingpolicy. Toanalyzenetworkstability,considertheLyapunovcandidate W(n):= o q ◦ (n) ⊤ IM ◦ (n)o q ◦ (n) withIM ◦ (n) being defined in (5.58). ThoughW(n) is indeed an energy function asIM ◦ (n) is pseudo positive definite by Lem. 5.1, due to the nonsymmetric weighting matrix IM ◦ (n), it has notrivialinterpretationofaspecificenergyinthesystem. Nonetheless,itclearlypenalizeshigh 152 MulticlassMinimumCostRouting queuedifferentialsacrosslinks,compellingamoreevendistributionofpacketsoverthenetwork. Italsoincitestransmissionoverthelinksoflowercostfactors,leadingtoalessexpensiverouting decision. Specifically, when all links and all classes are of the same cost factor, IM ◦ (n) is simplified to a scaled identity matrix, which in turn reduces W(n) to the sum of squares of queuelengths—afamiliarLyapunovfunctioninmostofpreviousresultsinliterature. Let ∆ W(n) := W(n + 1)− W(n) be the Lyapunov drift. Substituting for o q ◦ (n + 1) from (5.5)anddroppingtimeslotvariable(n) forthenotationease,weobtain ∆ W =(o a ◦ − IB ◦ f f) ⊤ (IM ◦ +IM ⊤ ◦ )o q ◦ +(o a ◦ − IB ◦ f f) ⊤ IM ◦ (o a ◦ − IB ◦ f f). Exploitinginequality (5.60)with16η 63 yields ∆ W 6η (o a ◦ − IB ◦ f f) ⊤ IM ◦ o q ◦ +(o a ◦ − IB ◦ f f) ⊤ IM ◦ (o a ◦ − IB ◦ f f) Substitutingf f ⊤ diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ forf f ⊤ IB ◦ ⊤ IM ◦ o q ◦ usingequality (5.59),weget ∆ W 6η o a ◦ ⊤ IM ◦ o q ◦ + η 2 ∥diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ − f f∥ 2 − η 2 o q ◦ ⊤ IB ◦ diag(o ρ ) − 2 IB ◦ ⊤ o q ◦ +(o a ◦ − IB ◦ f f) ⊤ IM ◦ (o a ◦ − IB ◦ f f)− η 2 f f ⊤ f f. (5.63) Giventhecurrentqueuebacklogs o q ◦ (n),letf f ⋆ (n)bethelinkactualtransmissionsprovided by the Dirichlet-based policy and evaluate (5.63) at f f ⋆ (n). As compared to any alternative transmissiondecisionf f(n),Th.5.5secures∥diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ − f f ⋆ ∥6∥diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ − f f∥ ateachslotn. Onealternativetransmissiondecisionisthecasewheref f(n)isproducedbya routing algorithm which makes independent, stationary and randomized transmission decisions at each slot n based only on arrivals and link capacities and so independent of both queue backlogsandlinkcostfactors[72]. Letusfix f f(n)forsuchanalgorithmandrefertoitasf f ′ (n). UtilizingthisintheLyapunovdrift (5.63)anddoingsomealgebra,weobtain ∆ W 6η o a ◦ ⊤ IM ◦ o q ◦ − ηff ′⊤ diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ +Γ Γ:=( o a ◦ − IB ◦ f f ⋆ ) ⊤ IM ◦ (o a ◦ − IB ◦ f f ⋆ )+ η 2 f f ′⊤ f f ′ − η 2 f f ⋆⊤ f f ⋆ . In light of equality (5.59), let us replacef f ′⊤ diag(o ρ ) − 1 IB ◦ ⊤ o q ◦ byf f ′⊤ IB ◦ ⊤ IM ◦ o q ◦ . Taking condi- tionalexpectationgiventhecurrentqueuebacklogs o q ◦ (n) thenleadsto E ∆ W|o q ◦ 6η E o a ◦ ⊤ IM ◦ o q ◦ o q ◦ − η E f f ′⊤ IB ◦ ⊤ IM ◦ o q ◦ o q ◦ +E Γ |o q ◦ (5.64) 5.8SimulationResults 153 wheretheconditionalexpectationiswithrespecttotherandomnessofarrivals,channelstates androutingdecision—incaseofarandomizedroutingalgorithm. Sincethenetworklayerroutingcontrollerhasnoimpactonarrivals, o a ◦ (n)turnstobean independent system variable that is not influenced by either IM ◦ (n) or o q ◦ (n), which implies E o a ◦ ⊤ IM ◦ o q ◦ = E{o a ◦ ⊤ }E{M ◦ |q ◦ }. Further, observe that IM ◦ (n) is a function only of link cost factors o ρ (n), and so independent of o q ◦ (n), which impliesE{M ◦ |q ◦ } = E{M ◦ }. Consideringthatf f ′ (n) isindependentfrom o q ◦ (n) andIM ◦ (n),(5.64)leadsto E ∆ W|o q ◦ 6η E{(o a ◦ − IB ◦ f f ′ ) ⊤ }E{IM ◦ } o q ◦ +E Γ |o q ◦ . (5.65) InvestigatingΓ( n), note that (i) all arrivals are of finite mean and variance, (ii) each link actualtransmissionisatmostequaltothelinkcapacitywhichisfinite,andsoboth f f ⋆ (n) and f f ′ (n)havefiniteupperbounds,and(iii) IM ◦ (n)isapseudopositivedefinitematrixinthesense of Lem.5.1 with finiteentries (recall ρ ij (n)>1). Thus, theexpected value ofΓ( n) is finite at each slotn, and so there exists a finite positive scalar Γ max such thatΓ( n)6 Γ max . Plugging thisintotheLyapunovdriftinequality (5.65),weobtain E ∆ W|o q ◦ 6η E{(o a ◦ − IB ◦ f f ′ ) ⊤ }E{IM ◦ } o q ◦ +Γ max . (5.66) ThenthenexttheoremisprovenbyshowingthatE{∆ W|o q ◦ }isalwaysnegative. Theorem5.7. (ThroughputOptimality)Onmulticlasswirelessnetworks,theDirichlet-based routing policy is throughput-optimal in the sense that it guarantees network stability for any trafficmatrixinthenetworkcapacity region. Proof. The rest of the proof follows the same direction and steps as that for Th. 3.2 in Ch. 3 withtheonlydifferencethatuniclassarraysthereneedtobereplacedbytheircorresponding multiclasshyper-arrayshere. 5.8 SimulationResults ConsiderthesamesimulationtestbenchbuiltinCh.4with50nodes,128linksandconnectivity graphofFig.4.2. Wirelesslinksareconsideredtobetwo-waychannels,i.e.,foranydirected linkij∈E there existsji∈E with the same capacity and cost factor. The network runs under 1-hopinterferencemodel,i.e.,linkswithcommonnodecannottransmitatthesametime. 154 MulticlassMinimumCostRouting 0 2000 4000 6000 8000 10000 0 2 4 6 8 x 10 7 0 2000 4000 6000 8000 10000 0 1 2 3 4 5 x 10 7 0 2000 4000 6000 8000 10000 0 0.5 1 1.5 2 2.5 x 10 7 0 2000 4000 6000 8000 10000 0 2 4 6 8 10 12 14 x 10 8 0 2000 4000 6000 8000 10000 0 2 4 6 8 10 x 10 8 0 2000 4000 6000 8000 10000 0 0.5 1 1.5 2 2.5 x 10 8 7 10  timeslot timeslot timeslot 7 10  7 10  Dirichlet-based V-parameter BP Dirichlet-based V-parameter BP Dirichlet-based routing V-parameter BP Total queue congestion 1   5   10   8 10  8 10  8 10  timeslot timeslot timeslot Total routing penalty Dirichlet-based V-parameter BP V-parameter BP Dirichlet-based V-parameter BP Dirichlet-based 1   5   10   dead-band dead-band dead-band Fig.5.4Timeslotevolutionoftotalroutingcostandtotalnumberofpacketsinthemulticlassnetwork,displayingtheperformance of Dirichlet-basedpolicy versus V-parameterBPpolicy withV = 0.8 underthreearrivalrates ofλ = 1 (left),λ = 5 (middle) and λ = 10 (right) for each class at each node. Besides lower routing penalty, note the far smaller oscillations under the Dirichlet-based routing, which implies a smoother flow of traffic. Total number of packets in the network is the representative of routingdelay,wherethe Dirichlet-basedpolicy makes the best possibletrade-off on cost versus delay. 5.8SimulationResults 155 Everytimeslot,capacityofeachlinkfollowsaGaussiandistributionwiththemeanofm ij and thevarianceof150packets. Toassignm ij ,weadoptShannoncapacitywithpowertransmission P ij ,noiseintensityN ij ,andabandwidthof1500,viz. m ij =1500log 2 (1+P ij /N ij ). Eachnode can expend 30 units of transmission power per timeslot, which under 1-hop interference model leads toP ij =30 foreach activatedlink. We randomlyassign anoiseintensityNij∈[1,5] to eachlinkatthebeginningandkeepitfixedduringthesimulation. We let every node send packets to every other node, forming a multiclass multihop wireless network. Different classes are generated at each node following Poisson’s random variables with parameterλ , where all of them are i.i.d. over timeslots and over nodes. Each link receives a random cost factorρ ij ∈[1,10], which is considered the same for all classes, at the beginning andiskeptfixedduringthesimulation. WiththetotalroutingpenaltyR(n):= P ij∈E P d∈K ρ ij f (d) ij (n) 2 ,thetoppanelsinFig.5.4 display the timeslot evolution of R(n) for three arrival rates corresponding to the Poisson parameters λ = 1, 5, 10 packets per timeslot, comparing the performance of our routing protocol and V-parameter BP withV = 0.8. Note that the Dirichlet routing cost, as defined in (5.1), reads the expected time average ofR(n). Besides noticeably lower routing cost, the Dirichlet-basepolicyresultsinfarsmalleroscillationsinR(n),whichimpliesmorestability andsmoothnessinthepacketflow. Further,whileourprotocolshowsanimmediateacttothe traffic rate, V-parameter BP has a latency by waiting until the network reaches a minimum total queuecongestion,calleddead-bandhere,whichislargerthan8000KpacketsforV =0.8and growsevenlargerwiththeincreaseofV. ThebottompanelsinFig.5.4displaythetimeslotevolutionoftotalnumberofpacketsinthe networkforthethreedifferentPoissonparameters,whichhasdirectcorrelationwithnetwork delay by Little’s Theorem [83]. Observe that the Dirichlet-based policy not only minimizes the total routing cost, but achieves it with far smaller routing delay compared to V-parameter BP. In fact,asstatedbyRem.5.11,theaveragequeuecongestionunderourprotocolistheleastthat canbeobtainedsubjecttominimumroutingcost. Long-term average performance of the two routing policies are compared in Fig. 5.5 as a function of arrival rateλ changing between 1 and 10 packets in unit steps. The average is taken on the last 40000 slots, when the system runs for 50000 slots starting from zero initial condition. Forλ =1, average total number of packets under our protocol is only 312K packets, comparedwith29400KpacketsunderV-parameterBP;likewise,theDirichletroutingcost (5.1) isonly5100Kunitsunderourprotocol,comparedwith91000KunitsunderV-parameterBP. Thisenormousdifferenceinperformancegetsonlylargerbythegrowthofarrivalrates λ . 156 MulticlassMinimumCostRouting 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 x 10 7 Arrival rate of each class at each node (packets/timeslot) Average total queue congestion (packets) 7 10  Dirichlet-based routing V-parameter BP routing 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12 x 10 8 Arrival rate of each class at each node (packets/timeslot) Dirichlet routing cost 8 10  Dirichlet-based routing V-parameter BP routing Fig. 5.5 Expected time average performance of the Dirichlet-based policy versus V-parameter BP policy withV = 0.8underarrivalratesgrowingfromλ = 1toλ = 10. Long-termaverageroutingcost(top). Long-termaverageroutingdelay(bottom). Dashedlinesdisplaythirddegreepolynomialinterpolation. 5.9 ConclusionandFuture Research Wedevelopedanetwork-layerroutingprotocolformulticlassmultihopwirelessnetworksthat minimizes a general quadratic routing cost subject to throughput optimality. To do so, we formulatedtheleastquadraticcostasaDirichletproblembyproposinganovelmappingbetween communication networks and multi-charge nonlinear resistive networks. The resulting protocol actsdynamicallywithnoprescribedroutingpathinformation, andrequiresnoknowledgeof 5.9ConclusionandFutureResearch 157 arrivalstatisticsandtopologyprobabilities,whichmake itusefulontime-varyingmobileand ad-hoc networks. As a broader impact, this research could open a new way to analyze and optimize the long-term average behavior of processing networks using mathematical properties ofconductioncurrentoverelectricalnetworks. A very intuitive line of future research is to apply all the theoretical works on improving BP tofurtherenhancetheperformanceofourroutingpoliciesproposedinCh.3–5,includingLIFO service [54], packet ages [53], and adaptive redundancy [55]. Specifically, it is interesting to developacross-layeroptimizationtechniqueinthespiritofourproposedpolicies[66–69]. In thiscase,thenetworkstabilityispossibleonlyifaflowcontrolmechanismintransportlayer canproperlylimittheamountofpacketsenteredintothenetworklayer. Thus,newlyarriving traffic first enters the transport layer where a cross-layer controller decides on the amount of datareleasedintothenetworklayerateachtimeslot. Another line of research is to analyze the impact of network topology on capacity region as an important issue in wireless networking. We proved that under our proposed routing policies (i)anytraffic matrixinthenetwork capacityregionisstabilized, and(ii)thetime averageflow ofpacketsonthenetworkcomplywiththecombinatorialheatflow,orconductioncurrentfor that matter. Then using the combinatorial geometry tools of Ch. 2, the network capacity region can be correlated with the stability of heat kernel on a smooth manifold. While the stability of heatkernelisnoteasilytractable,thereexistsharpboundsonitasafunctionofaverageRicci curvatureonthemanifold. Ontheotherhand,bytheconceptofgraphcurvature,whichisgoing tobeintroducedinPartII,wealreadyhaveagraphtopologymeasurethatcomplieswiththe notionofcurvatureingeometry. Sincethemorenegativecurvatureingeometryleadstoalarger bound on the heat kernel, our conjecture is that the same pattern should happen on the graph as a discrete domain, meaning that the more negatively curved topologies should result in smaller network capacityregions. (Foran initialread,referto[104].) It would be beneficial to investigate whether our proposed routing policies can be improved bygreedyhyperbolicembedding[49]. Thisispreciselytheapplicationwheretheextraflexibility offered by the link cost factor ρ becomes quite relevant. BP with greedy embedding protocol is drivenbythelinkqueue-differential,thelinkcapacity,andaspecializedweightcomingfrom the embeddingG→H. Our protocols, on the other hand, are driven by link queue-differential, link actual transmission, and a link cost factor ρ . One could certainly adjust the ρ so that it becomesthecostofgoingalongthewronglinkinagreedyprotocol. PartII PowerGrids Chapter6 CombinatorialGraphCurvature TheunderlyinggraphofanetworkcanundersomeconditionsbeapproximatedbyaRiemannian manifold, and itcanbe said, with some cautions, that the graph curvature is the curvature of the embedding manifold. The twist brought about by Gromov is that the interpolating manifold can be dispensed of, as the so-called thin triangle condition allows us to determine whether the graphis negatively curved in the verylarge scale fromits mere combinatorialand distance properties. ThentheconceptofscaledGromovpropertyallowsittoholdinsomemediumscale, andbythesametokenextendingtheconcepttopositivelycurvedgraphs. Theimportanceof the curvature stems from the fact that it is customary to transfer commodities across a network inanoptimalfashion. Assuch,commoditiesfollowthegeodesicflowanditisafundamental Riemannian geometry paradigm that the geodesic flow is regulated by the curvature. More specifically, in a negatively curved network, there will be congestion areas, where the majority ofthegeodesicstransit. 6.1 CurvatureFlowon Riemannian Surfaces Ingeneral,theaimofgeometricflowsistosmoothlydeformgeometricmanifoldsintocanonical forms. Manygeometricflows,includingtheRicciflowandmeancurvatureflow,demonstrate similarsmoothingpropertiesastheheatequation. Herewegiveanextremelybriefreviewon thisveryinterestingsubjectandreferthereaderto[105,106,36,35]formoredetails. 162 CombinatorialGraphCurvature 6.1.1 RiemanniansurfaceandGauss-Bonnettheorem Suppose(S,g)isaRiemannian2-manifoldembeddedinR 3 ,calledRiemanniansurface. Let usdenotepointsinR 3 by(u,v,w)andpointsinR 2 by(x,y)andassumef(x,y):R 2 →S is a regular parameterization ofS. Regularity means that the partial derivativesf x andf y are linearlyindependentforanypoint(x,y)inthedomainoff,andhencespanthetangentplane ateachpoint. Equivalently,thecrossproductf x × f y isanonzerovectornormaltoS ateach pointandwehaveafieldofunitnormalvectorsas n f =(f x × f y )/∥f x × f y ∥. Thesquareofthelineelementds isdeterminedfrom ds 2 =g(df,df)=g 11 dx 2 +2g 12 dxdy+g 22 dy 2 andtheareaofthesurfaceS isdeterminedfrom A(S)= Z M q det[g ij ]dxdy. The Riemannian metric is also called the first fundamental form, which determines intrinsic properties of the surface that are independent of embedding. The extrinsic properties are determined bythe second fundamentalformh that returnsthe projectionof the secondpartial derivativesoff ontothen f field,viz., h 11 =f xx n f , h 12 =h 21 =f xy n f , h 22 =f yy n f . Given the first and second fundamental forms together, a manifold can uniquely be determined uptoarotationandtranslationinR 3 . GivenaRiemanniansurface,continuousfunctionG:f →n f thatmapsthesurfacetothe unit sphere is called Gauss map. The differential of the Gauss map W : df → dn f defines a type of extrinsic curvature, known as shape operator. The shape operator determines the directions in which the surface bends at each point. The Weingarten equations associate the shapeoperatortothefirstandsecondfundamentaltensorsby W =[h ij ][g ij ]. The eigenvalues of the shape operator correspond to the principal curvatures and the eigenvectorscorrespondtotheprincipaldirections,whicharethedirectionsoftraversingonthe surfacetohavemaximumandminimumcurvature. Thehalfofthetraceoftheshapeoperator gives the mean curvature, whichis the average ofthe principal curvatures. The determinantof 6.1CurvatureFlowonRiemannianSurfaces 163 the shape operator, on the other hand, determines the Gaussian curvature, which is equal to the productoftheprincipalcurvatures. TheremarkablefeatureofGaussiancurvatureisthatwhilebydefinitionitcertainlydepends ontheembedding,itisanintrinsicinvariantdeterminedsolelyusingtheRiemannianmetric. Thegeodesiccurvatureisalsoanintrinsicinvariant. AsweknowfromCh.2,asmoothmanifoldcanbeequippedwithinfinitelymanyRiemannian metrics. While differentmetricsinduce different curvatures at each point, the Gauss-Bonnet theorem states that the total curvature is solely determined by the topology of the manifold and independent of the metric, implying that the metric has a topological constraint. SupposeS is a compact Riemannian surface with possible boundary∂S. LetK(g) be the Gaussian curvature ofS andκ (g) bethegeodesiccurvatureof∂S,bothinducedbymetricg,then Z M K(g)dA+ Z ∂M κ (g)ds=2πχ (S) wheredAistheelementofareaofS,dsis theline element along∂S, andχ (S) is theEuler characteristic ofS. Thetheoremconnects,somewhatsurprisingly,thegeometryofasurface induced byg to its topology inducedbyχ . Forasurfaceχ (S)=2− 2genus(S)− b(S) withb denotingthenumberofboundarycomponents. 6.1.2 Conformalequivalence TwoRiemannianmetricsg and ¯g onasmoothmanifoldMarecalledconformallyequivalent if ¯g = ug for some smooth function u :M→ R >0 that measures the area distortion and is calledconformalfactor. Thesetofall conformallyequivalentmetricsformaconformalclass onM. A diffeomorphism between two manifolds is called a conformal map if their metrics areconformallyequivalent. Conformal maps preserve both angles and shapes of infinitesimal subspaces,butnotnecessarilytheirsize. ForanyRiemanniansurfacethereexistsaconformalparameterizationforwhichtheRie- mannian metric is conformal to the Euclidean metric. The related local coordinates are called isothermal coordinates in which the Riemannian metric has a diagonal form with g 11 = g 22 = u(x,y). In isothermal coordinates, the Gaussian curvature takes on the sim- ple formofK(x,y)=− ∆ M u(x,y) where∆ M = ∂ 2 x +∂ 2 y /u denotes the Laplace-Beltrami operatorinducedbyg (seeCh.2). 164 CombinatorialGraphCurvature Given a compact Riemannian surface, the uniformization theorem states that any conformal class has a canonical metric which induces a constant Gaussian curvature λ ∈ {+1,0,− 1} everywhereaccordingtotheEulercharacteristic. Furthermore,thecanonicalmetricsglobally minimizetheenergyfunctional E(g):= Z M K(g)dA(g) 2 whereK(g) anddA(g) denotethe Gaussiancurvature andareaelement inducedby themetric g. One can confirm that the energy functional E(g) is convex with unique global optima. Therefore, any closed surface can be conformally deformed to a canonical surfaceK 2 by its canonicalmetric,whereK 2 isequivalenttotheunitsphereS 2 forgenuszerosurfaces(χ> 0), the Euclidean planeE 2 for genus one surfaces (χ = 0), and hyperbolic spaceH 2 for high genus surfaces (χ< 0). Equivalently, using the canonical metric, one can embed the universal coveringspaceofanyclosedsurfaceinoneofthethreecanonicalsurfacesS 2 ,E 2 andH 2 . 6.1.3 SmoothsurfaceRicciflow The Ricci flow is an efficient way to compute the canonical metrics of a smooth manifold. It is an intrinsic geometric flow that deforms the Riemannian metric in a way analogous to the heat diffusion. It can also be viewed as a gradient flow which tends to improve irregularities in themetricbyexpandingnegativelycurvedregionsandcontractingpositivelycurvedregions. Given a Riemannian manifold(M,g), letg be a function of time variablet. Then the Ricci flowisdefinedbythenonlinearpartialdifferentialequation ∂ t g(t):=− 2Ric(g(t)) whereRicdenotestheRiccicurvature. Thefactor-2canbechangedtoanynonzerorealnumber by rescalingt. The minus sign ensures that the Ricci flow is well defined for t>0, mimicking thefactthattheheatequationcanrunforwards,butnot(usually)backwards,intime. On a surface, the Ricci curvature takes on the simple form ofRic(g)=gK(g) whereK(g) istheGaussiancurvatureinducedbythemetricg. Thus,the surfaceRicciflow isstatedas ∂ t g(x,y,t)=− 2K(x,y,t)g(x,y,t) whichcoincideswiththeYamabeflow onaRiemannian2-manifold. 6.1CurvatureFlowonRiemannianSurfaces 165 It is proven that for a closed surface with any initial Riemannian metric, the solution to the Ricci flow exists and is unique for all t ≥ 0, and if the total area preserved during the flow, the solution converges exponentially fast to a constant curvature metric conformal to the initial metric. TheproofsweregivenbyHamilton[107]andChow[108]forsurfaceswithnon-positive andpositiveEulernumbersrespectively. To see how the Ricci flow is indeed a kind of nonlinear diffusion equation, consider the isothermal coordinates with the metric tensor ds 2 = exp(2u(x,y))(dx 2 + dy 2 ). Then the Gaussiancurvaturetakesontheformof K(x,y)=− ∆ M u(x,y), ∆ M =e − 2u(x,y) ∂ 2 x +∂ 2 y . Bysomeelementarymanipulation,onecanthenwritetheRicciflowas ∂ t u=∆ M u which is manifestly analogous to the heat equation ∂ t T = ∆ T with ∆ = ∂ 2 x +∂ 2 y being the classicalLaplacianintheEuclideanmetric. Remark 6.1. Whilethe heat equation isalinear partialdifferential equation,the Ricciflowis indeedanonlineardiffusionequationbecauseoftheLaplace-Beltramioperator. SolutiontothesurfaceRicciflowconformallydeformstheoriginalmetricas g(x,y,t)=u(t)g(x,y,0), u(t)=exp − 2 Z t 0 K(x,y,τ )dτ . SincetheRicciflowdoesnotpreservearea, oneusuallyneeds tonormalize theflow such that R S dA(x,y,t) = constant. Otherwise, instead of evolving a given surface into its canonical form,wemightjustshrinkitssize. OnecanalsoevolveagivensurfaceintoaprescribedformbymodifyingtheRicciflow as ∂ t g(x,y,t)=− 2 K(x,y,t)− ¯ K(x,y) g(x,y,t) where ¯ K denotesthetargetcurvatureconsistentwiththeGauss-Bonnettheorem. 166 CombinatorialGraphCurvature 6.2 GraphCurvaturein the Local This section aims to provide a connection between the notion of curvature in smooth geometry andcombinatorialgeometry. Fromadiscretizationandgeometricpointofview,thenotionof cellcomplexprovidesanaturalwaytothis. Thoughthiscouldberatherrestrictivefromapurely graph-theoreticpointofview,aswelearnedfromPartI,itallowstoprovefinerresults. We consider an undirected graphG = (V,E) with the set of vertexes or nodesV and the setofedgesE. Leteachedgee∈E beendowedwithapositivelengthℓ(e),whichrepresents the cost of traversal viae. The length can be chosen to be uniformly 1 or adjusted to reflect geographicaldistance,radiobandwidth,channelquality,transmissiondelay,electricalresistance, traversalcost,amongothers. We make the following assumptions about the graph under consideration: (1) The graph is finiteandundirected. (2)Thegraphisconnected. Whenitisnotconnected,theneachconnected graph may be treated separately. (3) Every edge is the least cost path between its two endpoints. (4) There are no nodes of degree one or two. (5) There are no null homotopic loops formed by atmosttwoedges. We assume that G can be embedded in an oriented topological surface S without self intersections, meaning that the nodesV are identified with points in S and the edgesE are identified with simple curves connecting their ends without intersecting eachother. We further assumethattheembeddingislocallycompact,whichimpliesthatthegraphislocallyfinite,and soeachnodehasonlyfinitelymanyneighbors. 6.2.1 PiecewiseLinearGeometry As we discussed in Ch. 2, a cell complex is a collection of finite-dimensional vector spaces ofn-cells,whereann-cellisdefinedasasetofpointshomeomorphictoaclosed n-ball. An n-cell can be represented as an ordered set of nodes comprising a convex n-polytope. The naturalanalogyofann-dimensionalcellcomplexisapiecewiseflat n-manifold. Apiecewise flat manifold is a geometric manifold in the sense that it is endowed with a piecewise linear structureinmuchthesamewaythataRiemannianmanifoldisendowedwithadifferentiable linearstructure. Similarlywecandefinealengthfunctiontocalculatethedistancebyminimizing the lengths of piecewise linear curves. It can be proven that every piecewise flat manifold of dimensionuptothreecanbesmoothed,i.e.,endowedwithadifferentiablestructure[105]. LetG(V,E,F)representtheembeddedgraphcorrespondingtoa3-dimensionalcellcomplex with the set of 0-cellsV, the set of 1-cellsE, and the set of 2-cellsF. Being a cell complex 6.2GraphCurvatureintheLocal 167 entails that (1) every edge is included in two faces, (2) every two faces are either disjoint or intersectinonenodeoroneedge,and(3)everyfaceishomeomorphictoacloseddisc. Thus, there exists a Riemannian 3-manifoldM such that the graphG is a cell-division forM and each3-cellofG canlifttoanembeddingintheuniversalcoverofM. Using this setting, piecewise linear functions which are naturally defined at nodes in G may be viewed as the analogues of smooth functions onM. Furthermore, analogous to Riemannian metricsonM,onemaydefine piecewiselinearmetricsinGwithsingularityinnodes. Essentially, a piecewise linear metric can be defined as a length function ℓ:E →R >0 such that each cell is embeddedintheEuclideanspaceasanondegenerateEuclideancellwithedgelengthsdetermined byℓ. Inparticular,ifthecellisatriangle,themetricℓmustsatisfythetriangularinequalitieson thethreeedges. Non-degeneracycanbeexpressedbythefactthatallcellsmusthavepositive volume. Given a piecewise flat metric ℓ, we may have different notions of curvature [ 109, 110]. The mostnaturaloneisconsideredasthecurvatureateach(n− 2)-cellofapiecewiseflat n-manifold, whichisequaltothedihedralangledeficitfrom 2π multipliedbythevolumeof(n− 2)-cell, possiblyfollowedwithanormalization. Itturnsoutthatwhileindimensiontwothecurvatureis concentratedatnodes,indimensionthreeitisconcentratedatedges. 6.2.2 Localcurvatureforplanargraphs Agraphiscalledplanarifitcanbedrawninsuchawaythatnoedgescrosseachother. Forevery planargraphthereexistsalocallycompactembeddingintoaRiemanniansurfaceS. Considera triangulationplanargraphG(V,E,F)endowedwithapiecewiseflatmetric ℓ,whereallfaces aretrianglesandthelengthofedgesaroundeachtriangularfacesatisfythetriangleinequality. ThenthereexistsaRiemanniansurfaceS forwhichG isatriangulation,andso(S,G,ℓ)defines apiecewiseflattriangulatedmanifoldofdimensiontwowiththemetric ℓ. Definition 6.1. Onametrizedplanar graph(G,ℓ) withpossibleboundary∂G,thegraph local curvatureK a atanodea isdefinedas K a :=    2π − P f abc ∈F θ bc a a / ∈∂G π − P f abc ∈F θ bc a a∈∂G. whereθ bc a istheinteriorangleofthef abc triangleattheapexa. 168 CombinatorialGraphCurvature ∑ angles > 180º ∑ angles < 180º ∑ angles = 180º Fig. 6.1 Intrinsic curvature measured by triangulation. A geodesic triangle is the region bounded by threegeodesicsthatmeetinthreepoints. DespitetheEuclideanspace(flatcurvature),inthespherical (positive curvature) and hyperbolic (negative curvature) spaces, the triangles cannot be made from a flat piece of paper without tearing or stretching. By the local Gauss-Bonnet Theorem, the integral curvature mayberegarded as the correction toπ ofthesumofinteriorangles. Themetricℓ solely determinesthe combinatorial curvature at each node from the cosine law,whereindifferentbackgroundgeometries,trianglessatisfydifferentcosinelaws: cosθ a =    (ℓ 2 ab +ℓ 2 ac − ℓ 2 bc ) (2ℓ ab ℓ ac E 2 (coshℓ ab coshℓ ac − coshℓ bc ) (sinhℓ ab sinhℓ ac ) H 2 whereforsimplicitywedenotedℓ ab =ℓ(ab),forthelengthoftheedgeab∈E. 6.2.3 Localcurvaturefornon-planargraphs To embed a non-planar graph in an oriented topological surfaceS without self intersections, the surface necessarily needs to carry some handles. The genusg of the graph is defined as the minimum numberofhandles requiredbysuchasurfacetoallowanembeddingwithoutedges crossing. The possibility of this embedding is assured by [111] for any finite graph, which also proves thatg≤ (|V|− 3)(|V|− 4)/12. Further, it is shown that any connected graph can be strongly embedded, also called cellular embedded, i.e., the graph partitions the genus-g surface intoacollectionofcellsthataretopologicallyequivalenttodisks[112,Ch.3]. To make such a strong embedding, one may cut up the genus-g surface and unfold it in such a way that each node is surrounded by polygonal faces, from which the face angles can bedetermined. Thisway,itispossibletoextendthenotionofGaussiancurvaturedefinedfor planargraphstoanyfinitegraph. Thedifficultyinpracticeisthatforagivengraph,itisNP-hardtofindthesmallestgenus surface inwhichthe graph canbe strongly embedded [113]. The guiding idea is to think the 6.2GraphCurvatureintheLocal 169 graph as (quasi-)isometrically embedded inR n , in which case a cyclic ordering of a subset {ab k 1 ,··· ,ab km }∈{ab 1 ,··· ,ab deg(a) } of all edges attached to the nodea can be thought as anm-surface. In the extreme case wherem = n, a cyclic ordering of all edges makes up an n-surfaceinwhichthewholegraphcanbeembedded[112]. Itis,however,preferabletoembedthegraphinaspaceofhigherdimension,asthisleaves moreroomtomaneuver,thereby makingthedistortionsmaller[114]. Thisforcesustochoose theotherextreme, thatis, tothinka surface tobe a cyclic orderingof aminimum numberof edges that is clearlym = 3. In this extreme case, a permutation of the ordering of the nodes is no more than a reversal of the ordering, where a subset of three edges{ab k 1 ,ab k 2 ,ab k 3 }, disregarding ordering, makes a sectionσ . Hence, the sectional curvature at nodea relative to a sectionσ isobtainedby κ (a,σ ):=2π − X 3 i=1 θ k i . Thenthegraphlocalcurvatureatnodea canbedefinedas K a := deg(a) 3 − 1 X σ ∋a κ (a,σ ). One can confirm that the above definition of curvature at a node is consistent with local clustering coefficient for a weighted graph, which is defined as the fraction of total value of trianglesoutoftotalvalueoftripletswithaasanode. 6.2.4 Fromgeometry(local)totopology(global) Theangledeficit/excessbaseddefinitionofcurvatureutilizesalocalapproachtographproperties, while many practical problems deal with the global performance of the network. Similar to the smooth case in Riemannian geometry, the local-to-global passage is addressed by the combinatorial Gauss-Bonnet theorem, stating that the total graph curvature is a topological intrinsicproperty,independentofthelocalcurvatureateachnode,whichsatisfies X a∈V K a =2πχ (G) whereχ (G):=2− 2g denotesthe Eulercharacteristicofthegraph. In particular case of polyhedra, we getχ (G) =|V|−|E| +|F| = 2 withF being the set offacesofthegraph. Bythesametoken,theEulernumberofeveryplanargraphembeddable on asphere is equal to 2. Notethat while for any planar graph|V|−|E| +|F| = 2 is always true, the graph’s Euler number is determined from its embedding surface. Thus, the Euler 170 CombinatorialGraphCurvature numberofcompletegraphK 7 ,embeddableonatorus,alsotheEulernumberofasquarelattice, embeddableonaEuclideanplane,arebothequalto0,whiletheEulernumberofX 3,7 tiling, embeddableonaPoincarédisk,isnegative. Definition6.2. InthesenseofGauss,agraphG isnegativelycurvedorhyperbolic,ifχ (G)<0; itisflator Euclidean, ifχ (G)=0;anditispositivelycurvedorspherical, ifχ (G)>0. 6.2.5 Reliabilityoflocalcurvature Despitetheappealofgraphcombinatorialcurvature,weprovideacoupleofsimpleexamples showingthatoneneedstobecarefulwiththeperformanceanalysisofanetworkbasedonits combinatorialcurvature. Lackoftriangulation: Considerasquarelatticewiththeunitlengthoneachedge. Whilethis graphisobviouslyEuclidean,theAlexandrovequationresultsintheangledeficitof − 2π ateach node that misleads us to a hyperbolic graph. This clearly shows that facing triangulated graphs with likely limited number of holes, if the graph is not well-triangulated, the combinatorial approach to curvature based on the Alexandrov equation is not reliable. This problem is circumventedbytheHiguchiformula[115],butonlydefinedforthegraphsofunitlengthedges. Lack of hyperbolicity: Consider atriangular latticeon the Euclideanplane withzero Euler characteristic. Letusaddnewdiagonaledgesconnectingeachnodetoitsoppositenodes(its second nearest neighbors). To embed this graph on a surface, one requires to pull out one handle, thereby increasing the genus, for each edge crossing. Observe that the average Gaussian curvature of the new graph is negative from the combinatorial Gauss-Bonnet theorem. But despite thischange incurvature, the globalperformance of thenetwork, e.g., the flow of traffic along geodesics or the average length of a random walk, is unaffected at a coarse-grained level. Lackofplanarity: Intheprevious example, if the number of new crossing edges is small enoughcomparedtothesizeofthegraph,thencontrarytowhatthecombinatorialGauss-Bonnet theoremstates,theaveragecurvatureremainspositive. Thisshowsthatforalmostplanargraphs with limitednumber of localnon-planarity, insistingon the genusof the graphwith the Gauss- Bonnet theorem is not reliable. Perhaps a better way of extending the Gauss-Bonnet theorem to generalgraphsistothinkofthetotalcombinatorialcurvatureasatopologyinvariant,unaffected bythelocalgeometryduetothelengthsofedges. 6.3CurvatureFlowonPlanarGraphs 171 6.3 CurvatureFlowon Planar Graphs Inagiventriangulationplanarmetrizedgraph(G,ℓ),twopiecewiselinearmetricsℓ andℓ ′ are calledconformallyequivalent ifℓ ′ ab =u(a)ℓ ab u(b),∀a, b∈V,andforsomepositivefunction u : V → R >0 which is called combinatorial conformal factor. The set of all conformally equivalentpiecewiselinearmetricsformacombinatorialconformalclassinG. Acombinatorial conformal map is a piecewise linear function that conformally changes the piecewise linear metric inG. The combinatorial conformal map is an approximation of a smooth conformal map in the sense that if we look at a graph as a cell-division of a smooth manifold and keep refining itbycellsubdivision,thecombinatorialconformalmapinthegraphconvergestothesmooth conformalmaponthemanifold[116]. 6.3.1 Circlepacking The concept of angle is brought to the piecewise linear geometry with the notion of circle packing. Thurston in [36, 35] defined circle packingby three properties: (1) map infinitesimal circlestocircleswithfiniteradii;(2)eachcirclewithradius r a determinesanodeaatthecenter; (3)thereisanedgebetweenaandbiftheircirclesintersecteachother,wheretheintersection angleϕ ab determines the weight of the edge. Thurston proved that for any three non-obtuse angles and any three positive radii, in both Euclidean and hyperbolic geometries, there exists a configuration of three circles unique up to isometry that meet above definition of circle packing. The non-obtuse condition0≤ ϕ ab ≤ π/ 2 promises triangle inequalities on the triangle formed bythecentersofthethreeintersectingcircles. Notethatthestandardcirclepackingcomesfrom ϕ ab =0 wherecirclesaretangenttoeachotherwithnooverlapping. On atriangulationplanar graph, a circle packing metric is defined with a pair of positive functionsR :V →R >0 that assigns a radius to each node andΦ : E → [0,π/ 2] that assigns a weight to each edge. A circle packing metric(R,Φ) is a piecewise linear metric inG that uniquelydeterminesthelengthmetricℓ indifferentbackgroundgeometriesbythecosinelaw, ℓ ij =    q r 2 i +r 2 j +2r i r j cosϕ ij E 2 cosh − 1 coshr i coshr j +sinhr i sinhr j cosϕ ij H 2 . (6.1) By some elementary manipulation, two circle packing(R,Φ) and(R ′ ,Φ ′ ) are conformally equivalent if and only ifΦ ′ =Φ . Thus, a combinatorial conformal map in a planar graph solely changes thenoderadiiofacirclepacking,whilepreservestheedgeweights. 172 CombinatorialGraphCurvature a r a ab φ b c ab ac bc d Fig. 6.2 Circle packing on a triangulation planar graph inE 2 where the length metricℓ is determined fromacombination of circle radii and intersectingangles. 6.3.2 CombinatorialRicciflow Given a metrized planar graph (G,ℓ), a natural question is whether there exists a constant curvaturemetricinacombinatorialconformalclass,andifsowhatthatmetricis. Accordingto[117,Theorem1.1],allplanargraphswithnon-negativeEulercharacteristic have a constant curvature metric. Furthermore, there is a combinatorial obstruction that guaran- tees the existence of constant curvature metric for the graphs of negative Euler characteristic if andonlyifforanypropersubsetW ofnodes,theinequality|F W |/|W|>|F|/|V|holds,where F W isthesetofalltriangleswithanodeinW. Analogoustothesmoothgeometry,thisconstant curvaturemetric can be computed by combinatorialRicciflow. Theideais todeformthenode radiiinaccordancewiththecurvaturedeficitateachnode. ThecombinatorialRicciflowisin factaheatequationthatcanbemodeledasanegativegradientofaconvexenergyfunctionwith uniqueglobalminima. Theideaistodeformthenoderadiiinaccordancewiththecurvature deficitateachnode. ThecombinatorialRicciflowisinfactaheatequationthatcanbemodeled asanegativegradientofaconvexenergyfunctionwithuniqueglobalminima. GivenaplanargraphG togetherwithacirclepackingmetric(R,Φ) ,thecombinatorialRicci flowindifferentbackgroundgeometriesaredefinedas da(t) dt =    − K a (t)r a (t) E 2 − K a (t) sinhr a (t) H 2 . (6.2) 6.3CurvatureFlowonPlanarGraphs 173 Theorem 6.1. (Combinatorial Ricci Flow) Consider a circle packing metric (R,Φ) for a planar graph. Solution to the combinatorial Ricci flow (6.2) exists, in both Euclidean and hyperbolic geometries, for any initial metricr(0) ∈ R |V| >0 and for all t ≥ 0. Further, in the Euclidean geometry, r(t) converges exponentially fast to the metric of constant curvature K ave =2πχ (G)/|V| ateverynode,inagreementwiththeGauss-Bonnettheorem. Similartothesmoothsetting,wecanmodifythecombinatorialRicciflowtodeformagiven triangulationplanargraphintoagraphwithprescribedcurvatureateachnodea, dr a (t) dt =    − K a (t)− ¯ K a r a (t) E 2 − K a (t)− ¯ K a sinhr a (t) H 2 where ¯ K a denotes the target curvature at each node, consistent with the Gauss-Bonnet theorem. Thenthefollowingtheoreticalresultiscrucialinpractice. Theorem 6.2. (Curvature Mapping [118]) Consider thr mapping ψ : R → K that sends the vector of node radii r = (r 1 ,...,r |V| ) to the corresponding vector of node curvatures K = (K 1 ,...,K |V| ). In hyperbolic geometry, ψ is injective, meaning that the curvature determines its metric. In the Euclidean geometry, ψ is injective subject to normalized radii Q a r a =1,meaningthatthecurvaturedeterminesitsmetricuptoascalarfactor. ToobservethatthecombinatorialRicciflowisthegradientflowofaconvexenergy,let u a (t)=    lnr a (t) E 2 lntanh r a (t)/2 H 2 . TheninbothE 2 andH 2 geometries,thecombinatorialRicciflowtakesontheformof du a (t) dt =− K a (t)− ¯ K a . Accordingto[118,Lemma2.3],foranytwoarbitrarynodesaandbwehave(∂K a /∂r b )r b = (∂K b /∂r a )r a in the Euclidean geometry, and (∂K i a/∂r b ) sinhr b = (∂K b ∂r a ) sinhr a in hy- perbolic geometry. This is equivalent to the symmetric relationship ∂K a /∂u b = ∂K b /∂u a , implyingthatthedifferential1-form P a∈V K a du a isclosed(curlfree). 174 CombinatorialGraphCurvature Letu=(u 1 ,...,u |V| )bethevectorofconformalfactorsinacombinatorialconformalclass U anddefinethe combinatorialRiccienergy E(u)=− Z u u 0 X a∈V ¯ K a − K a du a . (6.3) ThenbyStokes’theorem,integral (6.3)ispathindependentforanyarbitraryinitialmetricin- duced byu 0 . Thus, we obtain∂E(u)/∂u a =K a − ¯ K a or equallydu a (t)/dt=− ∂E(u)/∂u a , implyingthatthecombinatorialRicciflowisindeedthenegativegradientflowof E(u). Specif- ically,theRiccienergyisminimizedbysuchmetricthatinduces ¯ K a ateachnode. Theorem 6.3. (Convexityof Ricci Energy[118]) In hyperbolic geometry, the combinatorial Ricci energyE(u):U →R is strictly convex. In the Euclidean geometry,E(u) is convex and satisfies E(u+c1) = E(u),∀c∈R. Further, the kernel ofE(u) is solely generated by the vector1,meaningthatE(u) is strictly convexwhenrestrictedto P a u a =0. 6.3.3 Conformalmapcomputation To parametrize a given triangulation planar graph onto the plane, we first need to compute a circle packing metric(R,Φ) which approximates the original induced Euclidean metric on the graphascloseaspossible. Basedon (6.1),onecanuseaconstrainedoptimizationmethodto compute(R,Φ) using X ab∈E ℓ ab 2 − (r a 2 +r b 2 +2r a r b cosϕ ab ) 2 , ϕ ab ∈[0,π/ 2] whereℓ ab =ℓ(a,b) denotesthethelengthoftheedgeab∈E. KnowingthattheRicciflowisthenegativegradient flow oftheRicci energy, itsuffices to minimizeE(u)usingthegradientdescentmethod,whichisthedirectanalogyofthesmooth Ricciflow. Theconvergencespeedcan furtherbeimprovedby usingNewton’smethod,where thekeyistocomputetheHessianmatrixaccordingtothebackgroundgeometry. IntheEuclidean geometry,theHessianmatrixofE(u) isobtainedas ∂ 2 E ∂u a ∂u b = ∂K a ∂u b =r b ∂K a ∂r b =            r b P c∈V M ab c √ 1− (P ab c ) 2 a=b 0 a̸=b & ab / ∈∂G r b P c∈V N ab c √ 1− (P ab c ) 2 a̸=b & ab∈∂G . 6.4GraphCurvatureintheLarge 175 wherethetermsM,N andP arecalculatedby M ab c =2r b r c (r a +2r b +r c )/(r a +r b ) 2 (r a +r c ) 2 N ab c =− 2r a r b 2 /(r a +r b ) 2 (r a +r c ) P ab c =1− 2r b r c /(r a +r b )(r a +r c ). 6.4 GraphCurvature in the Large Asanalternativetothecombinatorialcurvature,negativecurvatureofalargemetricspaceis definedbyGromov[119,120]intermsof δ –thintrianglecondition. Wedefinea pathp(s,t)betweentwoarbitrarynodessandtasasequenceofnodes,including sandt,withanedgebetweenanytwosuccessiveonesandwithnorepeatednodeinthesequence. Thelengthofapath,ℓ(p(s,t)),isdefinedasthesumofthelengthsoftheedgestraversedbythat path. A path betweens andt is a geodesic, denoted by[s,t], if its length is minimum compared to all other possible paths joinings tot. The geodesic distance betweens andt, denoted by d(s,t),isdefinedasthelengthofthegeodesicpath [s,t],meaningthatd(s,t):=ℓ([s,t]). Remark6.2. Thedistanceoperatordsatisfiesthetriangleinequalityin G,meaningthatd(s,t)≤ d(s,m)+d(m,t) foranym∈E. Hence,thepair(G,d) makeametricspace inG. Wedefinea geodesictriangleas△abc:=[a,b] ∪ [b,c] ∪ [c,a]. Consideracyclicordering of the set of edges attached to the nodea as{ab 1 = ab 1 ,ab 2 ,··· ,ab deg(a) }. The Alexandrov angleθ k atthenodeaofthegeodesictriangle△ab k b k+1 ,fork =1,··· ,deg(a),isgiven by θ k =cos − 1 d(a,b k ) 2 +d(a,b k+1 ) 2 − d(b k ,b k+1 ) 2 2d(a,b k )d(a,b k+1 ) . Theorem 6.4. (Gromov’s Thin Triangle) For any geodesic triangle△stv built on the three arbitrarynodess,tandvofagraph,thereexistssucha(minimal)δ >0thateachgeodesic[s,t], [t,v]and[v,s] iswithintheδ -neighborhoodoftheunionoftheothertwo. For any geodesic triangle△abc, let us choose an arbitrary forth nodem. We determine the distancebetweenmandall thenodeson[a,b],anddefine d(m,ab) asthesmallestoneamong them. Let D(m,abc) be the maximum of d(m,ab), d(m,bc) and d(m,ca). The fatness of a geodesic triangle,δ (△abc), is defined as the minimum D(m,abc) among all arbitrary nodes m. Then the hyperbolicity of graphG is determined by the maximum fatness of all possible 176 CombinatorialGraphCurvature Fig.6.3 Gromov’s thin triangle condition for a hyperbolic geodesic metric space: (left) Each geodesic is withinδ -neighborhood of the union of the other two. (middle) The local combinatorial curvature at each nodeofX 3,7 hyperbolictessellationisnegative. (right)Arandomgraphformedbyjoiningnodesthat arewithinaspecified hyperbolic distance. geodesictriangles: δ (G):= sup △abc inf m D(m,abc) Definition 6.3. Aninfinite graphG is Gromovhyperbolicifδ (G)<∞. The intuition is that a surface with thin triangles is symptomatic of negative curvature. An extreme example of a Gromov hyperbolic graph is a tree, as δ (T) = 0. Relaxing the latter condition toδ (T)≈ 0 leads to a fattened tree, which is another less dramatic example. Intuitively,a coreconcentricnetworkcancoarselybeviewedasafattenedstar. It is shown that if a graphG is negatively curved, then any graph quasi-isometric toG is alsonegativelycurved. Theprocessofreplacingametricspace,e.g.,agraph,byonewhichis quasi-isometric to it is called coarsening. Indeed, while the two spaces can have very different propertiesonasmallscale,theyretainthesamefeaturesinlargescale. Remark 6.3. We note that none of the graphs discussed in the previous section satisfy the thin triangleconditionandthereforetheyarenotnegativelycurvedinthesenseofGromov. For a finite graph, the constant δ is trivially finite. Consequently, for the thin triangle condition to be relevant for real-life graphs, it has to be adapted by requiringδ to be reasonably small compared to the size of the graph [121–123]. The idea is to compare the behavior of δ (△)/diam(△) in the standard Riemannian manifolds M κ of constant curvature κ . If 6.5CurvatureversusCongestion 177 diam(△)≥ R forareasonablybigR,thefollowingcanbeproven: sup △∈M κ< 0 δ (△) diam(△) < sup △∈M 0 δ (△) diam(△) = 1 3 < sup △∈M κ> 0 δ (△) diam(△) . Therefore,byafundamentalcomparisongeometryargument,onecansaythatthegraphG is scaledGromovhyperbolicif sup △∈G diam(△)≥ R δ (△) diam(△) < 1 3 . Thisinequalityallowsenforcementofnegativecurvatureatvariousscales. Itisshownthatat the very low scale of the nodes of a triangulated graph, this inequality is consistent with the resultsfromcombinatoriallocalcurvature,andalsowiththeHiguchiformula[115]. ResearchTopic: While the inequalityconditionofscaledGromovhyperbolicityprovidesa quantitativedefinitionofnegativecurvatureforfinitegraphs,therequirementthateverytriangle shouldbeinanegativelycurvedregioncanbeundulyrestrictiveinpractice. Furthermore,inthe limitofaninfinitelylargegraphwithinfinitelylargetriangles,itis possiblethattheinequality issatisfiedandyettheGromovhyperbolicitycondition δ (G)<∞isviolated[38]. 6.5 Curvatureversus Congestion Traffic onagraphisdrivenbyasimpletrafficmeasure,whichistherateof commodities tobe transmittedfromasourcestoatargett. Assumethatthetrafficroutsonthegraphinaleastcost fashion,i.e.,commoditiesaretransferredfromstotalongthegeodesic[s,t]. Then,themain connections between congestion and curvature in a network can be formulated by the following definitionsandtheorems. Definition 6.4. On a connected weighted graph, the moment of inertia with respect to a nodev isdefinedby φ (v):=(1/λ ) P w∈V (d(v,w)) θ ,forsomeconstantsθ > 1andλ> 0. Thenode withthe minimummomentofinertiaiscalledcentroid. Theorem 6.5. (Congestion WitnessofCurvature [36,35]) Consider a large, but finite, nega- tively curved connected graph, subject to uniformly distributed demand for commodities. Then, there are some specific nodes with very high traffic, which are the ones of least moment of inertia. Further, if the graph is non-negatively curved, then both the traffic and inertia are more evenlydistributedthaninthecaseofanegativelycurvedgraph. Further,ifthegraphispositively curvedwithenoughsymmetry,boththetrafficandinertiaareuniformlydistributed. 178 CombinatorialGraphCurvature R C Poincaré disk s t t  r C Br C (0)=U p q t'  t' s' R r s t s' p q B r (0)=U u t' Fig. 6.4 Traffic on Euclidean space (left) and hyperbolic space (right). The optimal paths (geodesics) are uniformlydistributedinaEuclideanspace,butmaximallydistributedatthecenterinaPoincaréspace. Theintuitionisthat,becausethegeodesicsinPoincarédiscarearched,theyspendmoretimeinthesmall hyperbolicball than in the small Euclideanball. Graphswithabsolutenegativeorpositivecurvatureare,ofcourse,theextremesituations. Thereallifenetworksaresomewherebetweenthesetwoextremes,withdifferentlocalcurvatures fromnegativetopositive. Inthiscase,amoredetailedcurvatureanalysisisrequiredtodetermine thecongestionpoints[30]. TheaimistoidentifyamanifoldM n anditsRiemannianmetricg,suchthatthedynamic nodes canbethought ofasoperating onthemanifold. Moreprecisely,thequestioniswhether there existsanisometricmapas(G,d)→(M n ,g). Ignoringallthemathematicalcomplexity, welookatthisproblemveryintuitively. LetnetworkG bealargeballB R (0)insomehyperbolic spaceH n ,representedasthetruncatedPoincarédiskinFig.6.4. Theroutingbetweenthesource s andthetargett isassumedtobeinanoptimalfashionforthehyperbolicmetric. In this model, the centroid is the origin of the ball B R (0). To quantify the maximum congestionoccurringatthecentroid,considerthetrafficloadinasmallball B r (0),r≪ R,as Λ t (B r (0))= Z B R (0)× B R (0) ℓ([s,t]∩B r (0)) dsdt where the integral is the total length of all traffic paths in the small ball, and as such it is a measureofthenumberofcommoditiesinB r (0). Itisarguedthat,nomatterhowtheoretical our model of the traffic load Λ t (B r (0)) is, it is remarkably accurate at confirming that the load atthecenter scalesasN 2 whereN :=|V| isthenumberofnodesinareal network[36,35]. 6.5CurvatureversusCongestion 179 To reconcile the differential geometric and experimental approaches, it suffices to show that, inanappropriatelydiscretizedversionofΛ t (B r (0)),forB r (0)⊂ B R (0)⊂ H n ,thelatterscales asN 2 . In[36,35],theasymptoticformulainH n isfoundas λ t (B r (0))= Λ t (B r (0)) vol(B R (0)) 2 =O(constant) where ‘constant’ means ‘independent of R.’ Thus, the traffic load Λ t (B r (0)) in a small ball vol(B r (0))nearthecentroidscalesasthesquareofthevolumeofthenetworkvol(B R (0)) 2 that is modeled as the truncated hyperbolic manifoldB R (0)⊂ H n . In a tessellation of the Poincaré disk by polygons of equal areas, a node can be associated with each polygon, andarea(B R (0)) becomesthetotalnumberofnodesN. Similarargumentscanbedevelopedinndimensions. Thus,ourmodelcorrectlypredicts thatthemaximumtrafficloadscalesas N 2 . Fora2-dimensionalEuclideanspace,itisprovenin[36,35]that λ t (B r (0))= Λ t (B r (0)) area(B R (0)) 2 =O constant R . Therefore,uptosomeconstant(˙ =),thetrafficloadatthecenterscalesas area(B R (0)) 2 R ˙ =area(B R (0)) 1.5 ˙ =N 1.5 . Moregenerally,inann-dimensionalEuclideanspace,wehave λ t (B r (0))= Λ t (B r (0)) vol(B R (0)) 2 =O constant R n− 1 . Hence,thetrafficloadatthecenterscalesas vol(B R (0)) 2 vol(B R (0)) (n− 1)/n ˙ =vol(B R (0)) 1+ 1 n ˙ =N 1+ 1 n . Withno doubt, asthe dimensiongets higher and higher, the Euclidean congestion decreases, andthegapbetweentrafficloadsinhyperbolicandEuclideanspacesincreases. Remark 6.4. While it is relatively easy to check the Gromov property of network, it is not easy at all to associate a dimension with a complex network. The remarkable feature is that the asymptotictrafficanalysis transcendsthedimension,atleastinnegativelycurvedspaces. Chapter7 NegativelyCurvedPowerGrids Motivated by the concept of the smart power grid, reliability of a power grid is investigated fromatopologicalviewpoint. InspiredfromRiemanniangeometryofmanifolds,itisclaimed that extreme load at specific parts of a large power grid can occur as a consequence of the local negativecurvatureinitshiddenmetricspace. Thisworkcontributestofourareas: (1)Itdrawsa newcourseinthetopologicalstudyofapowergrid,whichis,unlikemost previousstudies,in accordance with the electrical characteristic, not the topographical structure, of the power grid. (2) It extends the Riemannian geometry metaphor developed for data communication networks to the power grid, an area that has never pervaded before. (3) It develops a unifying approach to deal with power and data networks, precisely at a juncture where the smart grid is bringing the twonetworkstogether. (4)Itprovidesananalyticalmeasureforthecriticalityoftransmission lines and power stations in a bulk transmission system, which can find a place in reliability assessmentandcentralizedflowcontrolinthefuturesmartgrid. 7.1 Introduction Smart grid is a terminology with a disparate set of goals, which encompasses the entire electric powersystem,fromtheinitialsourcestothefinalconsumersofenergy. Abroadlyexpanded transmissionsystem,withlargequantityofstochasticrenewablepowerstations,istheconspicuity of the future smart grid. It means that the emerging transmission system should be resilient to larger disturbances and more distributed malfunctions, which means extreme reliability persistentlyremainsthemostimportantrequirementforthetransmissionsystem. Inamethodical view, the reliability of a complex system, like power grid, is defined as a function inversely 182 NegativelyCurvedPowerGrids proportional to the criticality of its subsystems, where the criticality of subsystems may be changed by modifying certain attributes. Then, the crucial problem is to find an appropriate measureforthecriticalityoftransmissionlinesandsubstationsinabulktransmissionsystem. Inadditiontotheclassicaldefinitionforthecriticalityofapowercomponentasafunction of its load and capacity, we claim that a deeper concept is the grid topology, say grid curvature, whichhighlyaffectsthecriticalityoftransmissionlinesandsubstations. Undersomeconditions, the graph of a communication network, or that of the power grid for that matter, can be approxi- mated by a Riemannian manifold. Then, the graph curvature can be defined as the curvature of its embedding manifold, where a fundamental Riemannian geometry paradigm implies that the geodesicflowisregulatedbythecurvature. As a bridge betweenmanifolds and graphs, the Gromov Thin TriangleCondition allows us to determine whether a graph is negatively curved in the very large scale. As dicussed in Ch. 6, thishasitsinceptioninapropertyofatriangledrawnonanegativelycurvedsurfacetohavethe sum of its angles less thanπ , giving it a thin appearance. Moreover, the new concept of scaled Gromovproperty [122]providesuswithanextensiontosomemediumscale,andbythesame tokentotheconceptofnon-negativelycurvedgraphsaswell. Theaimistoextendthecongestionanalysisdevelopedfornegativelycurvedcommunication networks to a similar phenomenon in the power grid. We construct an innovative resistive networkandinferthegeometryof powerflowinthetransmissionsystemfromthetopological structure of this resistive network. It is critical to understand that we investigate the topology of the power grid based on the electrical characteristic of power flows, while the existing literature is mostlybased onthe topographical structure ofthe power grid, which,as correctlyobserved in[124],isnotabletopredicttheelectricalbehaviorofthenetwork. Furthermore, some recent attempts have been initiated to make a connection between power gridtopologyandcascadebreakdownsleadingtomajorblackouts,usingsomeclassicalnetwork conceptssuchasclusteringandSmallWorld[125],anddegreedistribution[126]. Weclaimthat theRiemanniangeometryapproachtopowergridtopologyisabletoidentify,inananalytical way,somearchitecturepronetocreatecascadeblackoutsfromlocalfaults. Thedominoeffectin anegativelycurvedpowergridcanbeanalyzedinaRiemanniansetupasfollows: Ifafaultoccurs inaline,thislineisremovedfromthegraphrelatedtothepowergrid,withtheconsequencethat the resulting graph is even more negatively curved, and so exacerbating the congestion problem with inevitable further faults. Accordingly, we offer a curvature-based analytical method to measure the criticality of transmission lines and substations, which can be utilized in reliability assessment andcentralizedflowcontrol inthefuturesmartgrid[127]. 7.2DynamicModelofPowerSystem 183 j j + j 1 : + j − 1 + j 1 − + j j j Fig. 7.1Transmissionlinemodelinpowergrids: (left)Transformertapsettingrepresentation. (right) TransformerΠ -model. 7.2 DynamicModelof Power System We use the following notation in the power grid: N for the set of all substations (nodes) in the power grid;N(i) for the set of all nodes directly connected to nodei;B for the set of all directed branches with elements (i,j) as the sending-end node i and receiving-end node j; [G ij ],[B ij ],[G sh ij ],[B sh ij ]∈R |B| forthevectorofseriesconductance,seriessusceptance,shunt conductance,and shuntsusceptanceatallbranches,respectively;[n ij ]∈R |B| forthevectorof transformer tap settings at all branches with n ij = 1 when the branch (i,j) does not have a transformer;[B sh i ]∈R |N| forthe vectorofbus shuntsusceptance atall nodes;and[P Gi ],[P Li ], [Q Gi ],[Q Li ]∈R |N| for the vector of active power generation, active power load, reactive power generation, and reactive power load at all substations, respectively. We express the complex busvoltage atnodei asE i =V i exp(jθ i ) whereV i andθ i arethe voltagemagnitude and phase angle,respectively,andj:= √ − 1. 7.2.1 Multi-time-scalemodeling A power system consists of dynamics evolving in different time scales. Handling all these dynamicsinasinglemodelispracticallyinefficient,ifpossibleatall. Thus,duringfasttransients, the slow states are considered almost constant. Similarly, we assume that the fast transients are not excited during slow changes. The standard approach to analyze a multi-time scale system is singularperturbationthatiswrittenintheformof ˙ x=f(x,y) ˙ y =εg(x,y) wherex∈R n andy∈R m togethercomposethe(n+m)-dimensionalvectorofstatevariables, andε≪ 1 denotes the perturbation parameter. Letx s ,y s be the slow andx f ,y f be the fast components of the state variables, so thatx =x s +x f andy =y s +y f . Due to theε term, 184 NegativelyCurvedPowerGrids the dynamics ofy are faster than those ofx. Thus, for ε = 0 we get the quasi-steady-state approximationoftheslowsubsystem, explainedbythedifferential-algebraicsystem ˙ x s =f(x s ,y s ) 0=g(x s ,y s ). (7.1) Foragivenx s ,(7.1)isobviouslytheequilibriumconditionfory,butsincethisisachievedby ε=0,we couldstillhave ˙ y s ̸=0. Wedefine slowmanifold asaninvariantmanifoldonwhichthefastdynamicsarenotexcited, i.e.,x=x s ,y =y s . Thisn-dimensional manifold can be defined by m equationsy s =h(x s ) suchthatεh x f(x s ,h)=g(x s ,h)whereh x istheJacobianofhwithrespecttox. Inparticular, forε=0theslowmanifoldisdeterminedby (7.1). Oncetheslowmanifoldiscalculated,the slowdynamicsaregivenbythereducedorderslowsubsystem ˙ x s =f(x s ,h(x s )). In order to approximate the fast subsystem, we use the fact thatx is predominately slow, implyingx≃x s . Thus,thestatevariablesofthefastsubsystemareactuallythefastcomponents y f ,alsocalledoff-manifoldvariables, whicharefoundasy f =y− y s =y− h(x s ). Substituting thisin (7.1)weobtaintheapproximatefastsubsystemas ε ˙ y f =ε ˙ y− ε ˙ y s ≃g x s ,y f +h(x s ) . Notethatthelatterdefinesasystemwithequilibriumpointontheslowmanifold y f =0,and that the slow variablesx s are parameters for the fast subsystem. Linearizing the fast subsystem atapoint(x s ,h(x s ))ontheslowmanifold,weobtainε∆ ˙ y f =g y ∆ y f . Thisturnsoutthatthe stability of the off-manifold dynamics is determined by the Jacobian g y . It should be remarked that the slow variable cannot be considered constant during fast transients, when the initial conditionsarenotclosetotheslowmanifold. Asymptoticexpansiontheoremsguaranteethatafterextinctionofthefirstinitialtransient startingoutsidetheslowmanifold,thequasi-steady-statesubsystem (7.1)approximatestheslow dynamics of the original system withO(ε) if and only if, (i) the Jacobiang y is nonsingular; (ii) all disturbances remain in the region of attraction of the stable equilibrium of the off- manifold dynamics. Sincethe slowvariablesactasparametersofthefastsubsystem,duringa 7.2DynamicModelofPowerSystem 185 slowtransient,thefastdynamicsmayexperienceabifurcation. Atsuchpoints, thetime-scale decompositionbreaksdown,sincetheaboveassumptionsareviolated. 7.2.2 Time-scaledecompositionofpowersystems In the context of multi-time-scale dynamic systems, a general power system can be introduced bythefollowingthreemaincomponents. Power network: Transients of the network are of electromagnetic type which is very fast comparedtothetimeintervalofinterest. Thus,weassumeaninstantaneousresponseforthe network describedbyasetofsmoothalgebraicequations 0=g(x,y,z c ,z d ) wherey isthevectorofbusvoltageswiththesamedimensionofg. Short-term dynamics: The short-term time scale is associated with synchronous generators, induction motors, HVDC components and FACTS devices. The corresponding dynamics, also calledtransientdynamics,lasttypicallyforseveralsecondsfollowingadisturbance. Theyare capturedbyasetofsmoothdifferentialequations ˙ x=f(x,y,z c ,z d ) wherex isthecorrespondingstatevector. Long-term dynamics: The long-term time scale is due to phenomena, controllers and protectingdevicesthatacttypicallyoverseveralminutesfollowingadisturbance. Inthecase of controllers and protecting devices, they are generally designed to act after extinction of the short-term transients to avoid unnecessary actions or even unstable interactions with the short-term dynamics. Load recovery, voltage and frequency control, transformer load-tap- changer operation, shunt capacitor or reactor switching, and generator limitation control are some relevant components of long-term dynamics. The long-term dynamics are represented by acoupleofcontinuousanddiscrete-timeequationsas ˙ z c =h c (x,y,z c ,z d ) z d (k+1)=h d (x,y,z c ,z d (k)). 186 NegativelyCurvedPowerGrids By way of summary, we model the dynamics of a large scale power system with a set of differential-algebraic,continuous-discretetimeequationsas ˙ x=f(x,y,z c ,z d ) (7.2) 0=g(x,y,z c ,z d ) (7.3) ˙ z c =h c (x,y,z c ,z d ) (7.4) z d (k+1)=h d (x,y,z c ,z d (k)). (7.5) 7.2.3 Powerflowequations The net active powerP i and reactive powerQ i injections into substationi are determined by P i = P Gi − P Li and Q i = Q Gi − Q Li ,∀i ∈ N. For a given voltage profile and network configuration, P i andQ i arecalculatedfrom P i (E,θ,n )=V i 2 X j∈N(i) G ij +G sh ij +V i X j∈N(i) V j (G ij cosθ ij +B ij sinθ ij ) (7.6) Q i (E,θ,n )=− B sh i V i 2 − V i 2 X j∈N(i) (B ij +B sh ij )+V i X j∈N(i) V j (G ij sinθ ij − B ij sinθ ij ) (7.7) whereθ ij :=θ i − θ j . Theactiveandreactivepowerflowsinthebranchesarecalculatedfrom P ij (E,θ,n )=n 2 ij G ij V i 2 − n ij V i V j (G ij cosθ ij +B ij sinθ ij ) (7.8) Q ij (E,θ,n )=− n ij 2 (B ij +B sh ij )V i 2 − n ij V i V j (G ij sinθ ij − B ij cosθ ij ). (7.9) Thenthebranchapparentpowerflowisdeterminedby S ij (E,θ,n )= q P ij (E,θ,n ) 2 +Q ij (E,θ,n ) 2 . (7.10) Notethatinallequationsabove,phaseanglesappearthroughdifferencesonly,andsothey aredefineduptoanadditiveconstant. Thus,it isrequiredtotakeone bus,socalled slackbus, asthereferencebysettingθ slack =0. In practice, when the analysis of power system involves very slow transitions from one steady-state operating point to another, the system is modeled with all dynamics, including long-termones,atequilibrium. Onecanobtaintheequilibriumpointsforthedynamicmodel 7.3VirtualNetworksofPowerGrid 187 (7.2)–(7.5)byasetofalgebraicequationsas f(x,y,z)=0 , g(x,y,z)=0 , h(x,y,z)=0 whereforsimplicitywehavegroupedthediscreteandcontinuousvariablesinasinglevector z = (z c ,z d ) and defined h := (h c ,h d − z d ). This equilibrium model can be reduced by eliminatingsomeofthexandz variables. Onecankeepreducingthemodeluptothepoint wherenof andnohequationremains. Thiseventuallyleadstoareductionofequilibriummodel to the network equations only, which takes on the form ofg ′ (y,ω sys )=0. The latter describes the steady-state operation of a power system as viewed from a network perspective, which is composed of2|N ◦ | power flow equations (7.6) and (7.7) together with the phase reference equationθ slack =0 withN ◦ denotingthesetofallbusesbuttheslackbus. Definition 7.1. The network-only model is a set of2|N ◦ |+1 algebraic equations that model all dynamicsofapowersystematsomeequilibriumpoint. 7.3 VirtualNetworks of Power Grid Indatanetworks,orinclassicalnetworksingeneral,somecommoditiesaretransportedfrom some sources to some targett along a least cost (geodesic) path[s,t]. The cost of a path is defined as the sum of the weights of the edges traversed by the path. The weight of an edge could be defined in a way that reflects the actual physical cost incurred as the commodities borrowtheshared resourceℓ. Else,itcouldhavenophysicalmeaningandcouldbechosenso thatoptimalroutingyieldsawise utilizationoftheresources. The first difficulty with power grid compared to data networks is that the traffic is not expressed by such a simple variable as the rate of commodities passing through a node or a link. Instead,electricalpowerrequirestwovariablestobeidentified,ageneralizedcoordinate(charge) andageneralizedforce(voltage). Anotherchallengeisthatsuchacommodityaspackethasa specific header and is transferred from source to destination through an optimal path. However, electrical power, instead, flows along all transmission lines and substations from generating sourcetoconsumingloadsinaccordancewiththephysicsofpowerflow. Thetransportalongaleastcostpath[s,t]seemstodisqualifyclassicalnetworksasrepre- sentativeofthepowergridwherethepowerflowismultipath. Notquiteso! Simplistically,it could be argued that most of the power flows along the least resistive path [ 128, pg. 30]. But a betterargumentcanbemadefromtheLaplacian-admittanceofthebusmodel. Todoso,and 188 NegativelyCurvedPowerGrids alsotoemploythepowerfulmethodsofdifferentialgeometryinthepowerflowproblems,we introducethenotionofvirtualnetworks insuchawaythatthegeometryofpowerflowinthe powergridcanbeconcludedfromthegeometryoftrafficinthisvirtualnetwork. Then,ifthe virtual network is negatively curved, we expect to find some lines vulnerable to overloading, or some critical powerstationsinthegrid. Remark 7.1. Weintentionallyseparatepowerstationsfromtransmissionlines,sincecontrary to the routers in a data network, overloading hardly happens for the stations in a practical power grid;though,criticalpowerstationscanstillbeconsideredasgridsecurityweaknesses. 7.3.1 Intuitiveidea Given the network-only model of a power system at some operating point, we aim at composing a virtual network isomorphic to the power grid such that the change of power flow in the power grid obtained with respect to the present equilibrium point can be determined by electrical DC current in the resistive network. It is critical to understand that we do not concern ourselves about assigning uniform power flow to the whole power network, which is neither feasible noreconomical. Instead,weaimatuniformizingthesensitivityofdifferentpowerstationsor transmissionlineswithrespecttothechangesinthecurrentsteadystateconditions. The above concept of sensitivity introduces a measure of system robustness at a given operatingpoint,whichinturnenhancespowersystemsecurity. Practicallyspeaking,security analysisconsists inchecking thesystem abilityto undergoa listof specifieddisturbances with areasonableprobabilityofoccurrence,referredtoascontingencies. Anotherlevelofsystem security is preventive control, which refers to actions taken in the normal state to bring the systemfromaninsecuretoasecurestate. Withinthiscontext,ourapproachisclassifiedasa preventivecontrolpolicywiththeobjectiveofmaximizingsecuritymargins. 7.3.2 Virtualnetworks We make the following approximations about the network-only model: (1) loads are treated as constant power; (2) generators have constant voltage or constant reactive power; and (3) generatorsotherthantheslack-bushaveconstantactivepower. Thelattermeansthatanychange ingenerationleveliscompensatedbytheslack-bus,insteadofbeingsharedbyacertainnumber of generators through governor or load frequency control. In this case, theω sys variable appears only in the slack-bus active power equation, which can be solved separately for ω sys . This decreasesthenumberofequationsfrom2|N ◦ |+1 to2|N ◦ | inthenetwork-onlymodel. 7.3VirtualNetworksofPowerGrid 189 Giventhenetwork-onlymodelofapowersystem,anydisturbanceinthepowersystem,after extinctionoftransients,includinglong-termtransients,canbemappedtoachangeinvoltage magnitudes and phase angles at buses. Letη be some quantity of interest that can be expressed as a function ofU = (V,θ ) whereV andθ denotes the vector of voltage magnitudes and phase angles at all buses, respectively. If we assume that the disturbance is small enough not to bringthesystemintoinstability, the system will generally operate at some other equilibrium point. As a result,η will also change. For small changes inU, we are interested in determining thesensitivityofη toeachU i definedas S U i η = lim ∆ U i →0 ∆ η ∆ U i . (7.11) Differentiating η (U) yields dη = dU ⊤ ∇ U η . Hence, the sensitivity (7.11) is in fact the i-th componentofthevectorofsensitivitiesS U η =∇ U η . Basedonphysical propertiesofthepowergrid,operatingatsomeequilibriumpoint,there isastrongdependencybetweenactivepowerandbusphaseanglesononehand,andbetween reactive power and bus voltage magnitudes on the other hand. This in turn implies a weak sensitivity of active power to voltage magnitudes, and also a weak sensitivity of reactive power tovoltage phaseangles. More specifically, we get S V P ij =∇ V P ij ≈ 0 andS θ Q ij =∇ θ Q ij ≈ 0. From (7.8),ontheother hand,P ij dependsonlyonθ i andθ j ,andfrom (7.9)Q ij depends only onV i andV j ,whichresultin dP ij = ∂P ij ∂θ i dθ i + ∂P ij ∂θ j dθ j (7.12) dQ ij = ∂Q ij ∂V i dV i + ∂Q ij ∂V j dV j . (7.13) As an inherent characteristic of a practical power network, for most transmission lines, the lineresistanceismuchsmallerthanthelinereactance,whichinturnimpliesG ij ≪ B ij . Further, we assume sin(θ i − θ j ) ≈ (θ i − θ j ) and cos(θ i − θ j ) ≈ 1, and V i ≈ V j , which are justified bythefactthatinasteadystateequilibriumpoint,twoincidentbuseshaveasmalldifference in their phase angles and voltage magnitudes. One can also approximate B sh ij as a constant powerloadandsoaddituptoQ L . Thensomeelementarymanipulationprovidesuswiththe 190 NegativelyCurvedPowerGrids approximationsofsensitivityequationsof (7.12)and (7.13)as dP ij =(B ij V i V j )(dθ i − dθ j ) (7.14) dQ ij =(2B ij V i − B ij V j )dV i − (B ij V i )dV j . (7.15) Let us identifydP ij with a current anddθ i − dθ j with a voltage drop in (7.14). Then (7.14) canbeinterpretedasthevoltage-currentcharacteristicoftheso-calledvirtualP-resistivenetwork. Further, applyingV i ≈ V j , one can approximate (7.15) withdQ ij =(B ij V i )(dV i − dV j ) which makes the same interpretation possible for the so-called Q-virtual network by identifyingdQ ij withacurrentanddV i − dV j withavoltagedropalongatransmissionline. Definition7.2. Giventhenetwork-onlymodelofapowersystemataspecificequilibriumpoint, its P-virtual network is defined as a resistive network isomorphic to the power grid such that r ij =1/(B ij V i V j )foreachtransmissionlineij,voltagesatthenodesrepresentbusphaseangles, andcurrentsovertheedgesrepresentlineactivepowers. Definition 7.3. Given the network-only model of a power system at a specific equilibrium point, its Q-virtual network is defined as a resistive network isomorphic to the power grid such thatr ij =1/(B ij V i )foreachtransmissionlineij,voltagesatthenodesrepresentbusvoltage magnitudes,andcurrentsovertheedgesrepresentlinereactivepowers. NotethattheedgeresistanceinQ-virtualnetworkisdefinedbasedonthevoltagemagnitude atthesending-endbusi. Becauseouranalysisofpowersysteminvolvesveryslowtransitions from one steady-state operating point to another, the direction of reactive power is not changing between two successive equilibrium points. Thus,r ij =1/(B ij V i ) can be determined for every transmissionlinewithnoambiguity. ApplyingV i ≈ V j ,wecanreconfigure (7.15)basedonnormalizedchangesinbusvoltage magnitudesasdQ ij =(B ij V i V j )(dV i /V i − dV j /V j ). Consideringapparentpowerflow S ij = P ij +jQ ij throughatransmissionline,weobtain dS ij =(B ij V i V j ) (dθ i +jdV i /V i )− (dθ j +jdV j /V j ) . (7.16) Let us for each transmission lineij associatedS ij with a complex current and(dθ i +jdV i /V i ) with a complex voltage. Then (7.16) can be interpreted as the voltage-current characteristic of a resistive network with edge resistancer ij ∈R, complex nodal voltagesU i ∈C and complex edgecurrentsI ij ∈C. AlongwithP-andQ-virtualnetworks,thismaybereferredtoasS-virtual network ataspecificequilibriumpointofthepowersystem. 7.3VirtualNetworksofPowerGrid 191 Definition7.4. Giventhenetwork-onlymodelofapowersystemataspecificequilibriumpoint, its S-virtual network is defined as a resistive network isomorphic to the power grid such that r ij = 1/(B ij V i V j ) for each transmission lineij, voltages at the nodes represent the complex value(dθ i +jdV i /V i )foreachbusi,andcurrentsovertheedgesrepresentlineapparentpowers. Itisimportanttounderstandthatallvariablesofthesystem,whenevaluatedinaspecific equilibrium point, are assumed constant in short-term analysis, even though their values can change in medium- and long-term operation. Also notice that the root of fluctuations in the system is the variation of power supply/demand, happening at buses. Then, deviation of voltage magnitudedV i andphaseangledθ i willbetheconsequenceofthosevariations. Accordingly, thecorrespondingvirtualnetworkswillonlyhavecurrentsourcesintheirnodesasmetaphors forthefluctuationofbuscomplexpowers. Thestationaryvalueofeachcurrentsourceiszero, correspondingtonopowerfluctuationinthebussupply/demand. Thentheupshotisthenext theoremthatcorrelateslinepower fluctuationswithbussupply/demandvariations. Theorem 7.1. (Resistive Network versus Transmission Grid) Consider a virtual network associatedwithaspecificequilibriumpointofthepowersystem. Letcomplexcurrentsources ψ i injectintothenodes,resultinginnodecomplexvoltagesU i andedgecomplexcurrentsI ij . Then the fluctuation of line complex powers and bus complex voltages in the power grid respectively satisfydP ij +jdQ ij =I ij anddθ i +jdV i /V i =U i for each transmission lineij, if and only if thefluctuationofcomplexpowerat eachbus i satisfies dS i =ψ i . Proof. ItdirectlyfollowsfromthedefinitionofvirtualnetworksandTellegen’stheorem. 7.3.3 Electricalversusgeodesicdistance ConsideraresistivenetworkR(V,E)whereeachedgeij∈E isendowedwitharesistorr ij >0. Itisknownthatthecurrentbetweentwonodesiandj willnotnecessarilyfollowthepathofleast resistanceR(i,j). The relevant concept, instead, is that of effective resistance which represents electrical distance. The effective resistance between two nodes is a measure of how electrically closetheyare,i.e.,R eff (i,j)issmallwhentherearemanypathswithlowresistancebetween nodesiandj. Thepracticalimportanceoftheeffectiveresistance,atleastfromtheviewpointof thisstudy,isthattheasymptoticbehavioroftheeffectiveresistance R eff (i,j)versustheshortest pathresistanceR(i,j) canprovideinformationaboutthecurvatureofresistivenetwork. Remark 7.2. On a resistive networkR, the effective resistance satisfies the triangle inequality, viz.,R eff (i,j)≤ R eff (i,m)+R eff (m,j). This implies thatR eff is a distance, which is different 192 NegativelyCurvedPowerGrids fromthegeodesicdistance(shortestpathresistance)R,unlessifRisatree. Hence,(R,R eff ) makeanewmetricspacedifferentfrom (R,R). On a resistive network, effective resistance between every two nodes can be calculated from thegraphDirichletLaplacianofCh.2. LettingL ◦ betheDirichletLaplacianmatrixwithrespect totheground,weget R eff (i,j)=    (L − 1 0 ) ii +(L − 1 0 ) jj − 2(L − 1 0 ) ij ifneitheri norj aregrounded (L − 1 0 ) ii ifj isgrounded. 7.4 GeometryofPowerTransmission TheaimofthissectionistoapplytheabstractconceptsoutlinedinCh.6totheproblemofpower congestioninpowergrids. Thecurvatureconceptsarebeingdevelopedinordertoanticipate whichline,ifany,islikelytooverload. Theintuitiveconceptisthatthepowergridwillhave congested spotsifitisnegativelycurved. TheGromov hyperbolic propertyenters the scenery in the sense that, should the virtual network be Gromov hyperbolic, the power grid is negatively curvedanditselectricalbehaviorisnearlyisometrictoacore-centric network. 7.4.1 Hyperbolicresistivenetwork A possible node pattern is that of a Euclidean lattice, in which the nodes of the graph are obtained by the action of a discrete group of translations from a single node; and every two nodes,linkedbyageneratorofthetranslationgroup,areconnectedbyalink, i.e.,aresistorin the resistive networksetting [129]. FamiliarexamplesofEuclidean latticesincludethesquare graph and the cubic graph among others. One of the topological features of Euclidean resistive networks, as stated by the following theorem, is that, remarkably, the asymptotic properties oftheeffectiveresistance(electricaldistance)differfromthoseoftheshortestpathresistance (geodesicdistance),anddependonthedimensionofthelatticeaswell. Theorem 7.2. (Euclidean Resistive Network [130]) We get R eff (i,j) = O R(i,j) on a 1- dimensionalEuclideanresistivestring,R eff (i,j)=O logR(i,j) ona2-dimensionalEuclidean resistivelattice,andR eff (i,j)=O 1 ona3-dimensionalEuclideanresistivelattice. Now, we turn our attention to the opposite situation, where the effective resistance and theshortestpathresistancehavethesameasymptoticbehavior,independentofthedimension. 7.4GeometryofPowerTransmission 193 GivenaresistivenetworkR,wedefinea geodesictriangle△ijk asatrianglemadeupofthe shortestpathresistancesR(i,j),R(j,k) andR(k,i). Definition 7.5. A resistive networkR(V,E,R) is Gromov hyperbolic if there exists a finite δ ,suchthateverygeodesictriangle△ijk hasaninscribedtriangle△mnpofaperimeternot exceedingδ ,meaningthatR(m,n)+R(n,p)+R(p,m)≤ δ . Intuitively,aGromovhyperbolicgraphlooks likeatreewhenviewedat adistance,where theconceptofviewingatadistanceisformalizedinlarge-scalegeometry,alsoreferredtoas coarsegeometry [120]. Thegeneralideaofcoarsegeometryisthatspaceswhichmaylocally beverydifferentcanstillbeverycloseonalargescale,andthatmanypropertiesofthemcanbe invariant under some coarse approximations, i.e., isometries up to some bounded distortion. As farasgeometrictechniquesareconcerned,wedonotcareaboutdistortion,providedthatitis uniformlybounded. Inotherwords,inaresistivenetworkR,theexact valueoftheeffective resistancebetweennodesisirrelevant;buttherelevantfactiswhetherthisresistanceisvanishing, finite,orinfinite,whicharecoarsegeometricinvariants. Definition 7.6. An embedding f : G → H of the graphG = (V G ,E G ,d G ) into the graph H = (V H ,E H ,d H ) is a quasi-isometry, if for every arbitrary nodes i and j inG, there exist constantsλ ≥ 1,ε≥ 0 andc≥ 0 suchthat 1 λ d G (i,j)− ε≤ d H f(i),f(j) ≤ λd G (i,j)+ε andeverynodeinH hasadistanceatmostc fromsomenodesintheimagef(G). Remark7.3. Intheprocessofreplacingaspacewithitsquasi-isometricimage,oneofthemost interestingfeatures thatremains invariantin large-scaleis negative curvature,i.e., if onespace isnegativelycurved,soistheother. Conjecture 7.1. (Quasi-Isometry to Tree) LetR = (V,E,R eff ) be a Gromov hyperbolic resistive network subject to a quasi-pole and a Cantor Gromov boundary. There exist finite constantsα ≥ 1 andβ ≥ 0,atreeT =(V ′ ,E ′ ,R ′ eff ) andanembeddingf :G→T such that 1 α R eff G (i,j)− β 6R eff T f(i),f(j) 6αR eff G (i,j), ∀i,j∈V G . (7.17) Conjecture7.1justifiesthataGromovhyperbolicgraph,subjecttosometechnicalconditions, isisometrictoatreeuptoaboundeddistortion. Inotherwords,Gromovhyperbolicnetworks 194 NegativelyCurvedPowerGrids are a mathematical idealization of core-centric, negatively curved graphs. At the same time, thoughagraphquasi-isometrictoatreeisGromovhyperbolic, theremaynotbeidentifiable extrapropertyofaGromovhyperbolicgraphthatcouldsecurequasi-isometrytoatree. Theorem7.3. (HyperbolicResistiveNetwork[116])LetR=(V,E)beaGromovhyperbolic resistivenetworksubjecttoaquasi-poleandaCantorGromovboundary. Foranyi,j∈V,we getR eff (i,j)=O R(i,j) . Moreprecisely,weget lim R(i,j)→∞ R eff (i,j) R(i,j) ∈[1/α,α ] whereα , as defined in (7.17), is the multiplicative distortion in the quasi-isometry between the graphanditsembeddedtree[131]. 7.4.2 Gridcurvatureversus lineoverload To be practical, let us assign to each bus in the power grid an operation risk factor ρ , which representstheriskofexperiencingpowerfluctuationinthebus. Thenthe expectingvalueof powerfluctuationforeachbusisdeterminedintermsofthevalueofitspowersupply/demandin a specific steady-state operating mode. It means that if, for example, a bus operates under 1000 MVAwithariskfactorof0.1,thetransmissionsystemmustbereliableagainst1000× 0.1= 100 MVApowerincrement/decrementinthatbus. Toextendcongestionanalysisdeveloped forcommunication networks tothat ofthe power grid,wehavehithertodealtwithtwodifficulties,i.e.,definingtrafficasasimplevariable,and identifyingthewayinwhichthistrafficisdispatchedfromsourcetodestination. Therearea coupleofmorechallengeswhichareillustratedinFig.7.2bycomparisonbetweencongestion in data networks and overload in power grids. The first contrast is that in data networks the limitationisontherouterswithbufferdrops,whereasinpowergridsthatthelimitationisonthe power lineswith overload trips. Thesecond contrast is thatin data networksboth send/receive and congestionoccur in the nodes, whereas in power grids that supply/demand occurs in the nodesbutoverloadinghappensinthetransmissionlines(links). To proceed, we need to introduce a distance between a line and a bus in the power grid. Although defining a notion of distance between these two dissimilar objects is mathematically controversial,wecandoitinaccordancewiththephysicsofthesystem. AssumecurrentI x is injected into the nodex in one of the virtual networks we previously defined. This increases the 7.4GeometryofPowerTransmission 195 Data Network Flow of Packets Power Grid Flow of Power Packets Send/Receive Buffering Packets Nodes Packet Drops at Routers Topological Tool: Node Centrality w.r.t. Nodes Power Suply/Receive Adjusting Voltage Phase & Magnitude Nodes Overloading at Lines Topological Tool: Line Centrality w.r.t. Nodes Fig. 7.2 Node congestion in data networks versus line overload in power grids. In a data network, limitation is on routers with packet drops, whereas in power grid limitation is on transmission lines with overload trips. In a data network both send/receive and congestion occur in nodes, whereas in power grid supply/demand occurs in buses (nodes)butoverloadinghappensinlines(links). voltagebetweennodesk andm inthenetworkby u km = (L − 1 0 ) kx − (L − 1 0 ) mx I x . Hence,thelinkkm receivesacurrentwiththevalueof i km = 1 R km (L − 1 0 ) kx − (L − 1 0 ) mx I x . Onecanview0≤| i km /I x |≤ 1asameasureofelectricalcloseness betweentheedge(trans- mission line)km and the node (power station)x. Then, we define weighted electrical centrality (inverseofinertia)foratransmissionlinekmasthesumoftheweightedclosenessbetweenthis lineandallnodesinthecorrespondingvirtualnetwork,viz., C km := 1 (N− 1)R km X x∈V (L − 1 0 ) kx − (L − 1 0 ) mx |E x |ρ x (7.18) where E x ∈ {P x , Q x , p P 2 x +Q 2 x } represents the bus net power depending on the virtual network of interest, the constant06 ρ x 6 1 represents the bus operation risk factor, and the constantN representsthenumberofnodesinthenetwork. 196 NegativelyCurvedPowerGrids Definition 7.7. Consider a power grid in a steady-state power flow condition, with the bus operationriskfactorsandvirtualnetworksassociatedwiththisequilibriumpoint. Thenfora user-specifiedconstant α > 1,wedefinethe normalizedmomentofinertia ofthepowergrid withrespecttoatransmissionlinekm as φ km := 1− C km max ij∈E C ij α . (7.19) Remark 7.4. The moment of inertia of each line is defined in accordance with a specific stationary operation mode of the power system. Therefore, it is constant only in short-term analysis,andmayhighlychangeinmedium-andlong-termoperation. While the local curvature of power grid can hardly be determined, if possible at all, Th. 7.3 provides a practical tool to check the hyperbolic property of the power grid using the notion of virtual network. Then the following corollary predicts the geometry of power flow in a hyperbolic grid in terms of the line moment of inertia, which can analytically be determined for eachtransmissionlineusing(7.19). Corollary7.1. (NegativelyCurvedPowerGrid)Considerapowergridinasteady-statepower flow condition, with the bus operation risk factors and virtual networks associated with this equilibrium point. If the corresponding resistive network is Gromov hyperbolic, then under uniform distribution of power fluctuations in supply/demand, the lines with least moment of inertiawillexperiencemostfluctuationsintheirtransmissionload. Ontheotherhand,ifthe virtual network is not hyperbolic, then all lines have nearly the same moment of inertia and the fluctuationofpowertendstobedistributedamongtheminanearlyuniformmanner. It must be borne in mind that a small moment of inertia implies higher vulnerability to uncertainchangesinthepowertransmittedbyatransmissionline,butnotnecessarilyleadingtoa lineoverload. Todeterminethelinevulnerabilitytooverloading,onealsoneedstoknowanother characteristicoftheline,so-calledlineutilization. Initssimplestform,thelineutilizationis definedas F km =|S km |/|W km |,where|S km |= p P 2 km +Q 2 km isthelineapparentpower,and W km denotestheratedcapacityofthe lineinvolt-amp(VA). Corollary 7.2. (ReliableTransmissionCondition)Consideranegativelycurvedpowergrid in the sense of Cor. 7.1. Then to have areliable power transmission under uniform distribution of power fluctuations in bus supply/demands, the highest free capacity must be allocated to thosetransmissionlineswiththelowestmomentsofinertia. 7.4GeometryofPowerTransmission 197 It is important to discriminate between the classical way of regarding high transmission line utilization and the approach proposed here. Indeed, if a line is in high utilization, the red flag is already raised, even in a traditional dispatch. However, the claim here is that for a transmission line with respect to which the power grid has a low moment of inertia, the red flag must be raisedinaquitelowerutilizationlevelcomparedtothatpracticedinthetraditionaldispatch. Suchatransmissionline,evenwithlowerutilization,maybeathigherriskofoverloadinginthe presenceofunpredicteddisturbanceinsupply/demand. Remark 7.5. In a negatively curved power grid, if the line utilization remains globally low for allpossiblescenariosofsupply/demand,alowmomentofinertiashouldnotbeaconcernfor line overload, i.e., reliability against supply/demand disturbance. Nevertheless, if the grid is quasi-isometric to a tree, the line failure could cut the service to many consumers, i.e., still unreliableagainsttopological,orstructural,disturbance. 7.4.3 Effectoffluctuationofrenewables Oneofthebig drawbacks of“goinggreen”withsuchrenewableresourcesaswindfarmsand photovoltaiccellsisthefluctuatingnatureofthepowertheydeliver. Sincetherenewablesare distributed across the grid, they will inevitably induce fluctuations in the power flowing through the power transmission system with potential for voltage instability. This fluctuating power can ofcoursebefoundbysolvingthepowerflowequations,whichwouldofcourserevealthe“stress points,”butwouldnotexplainwhythestresspointsappearatsomeparticularbusesorlines. OurDCpowerflowequationsdepictthepowergridasaresistivenetworkwherethefluc- tuatingcomplexpower ˜ S km flowingthroughtheline kmcanbeviewedasacomplexcurrent flowingthrougharesistorsubjectto acomplexvoltagedropof ˜ S km = ˜ P km +j ˜ Q km = B km ¯ V k ¯ V m cos ¯θ km ( ˜ θ k +j ˜ V k ¯ V k )− ( ˜ θ m +j ˜ V m ¯ V m ) . In the equation above, the tilde quantities represent fluctuations relative to the mean quantities denotedbyoverbar. Usingsomenetworkconcepts,itistheneasytocharacterizethestresspoint asthose ofhighbetweennesscentrality. 198 NegativelyCurvedPowerGrids 0 50 100 150 200 250 300 0 5 10 15 20 25 30 Electrical Inertia vs Transmission Power Node number in Virtual Network 0 50 100 150 200 250 300 350 400 450 0 2 4 6 8 10 12 14 Bus Transmission Power (100 MV A) Bus Moment of Inertia Electrical Inertia vs Transmission Power Link number in Virtual Network Line Transmission Power (100 MV A) Line Moment of Inertia Fig.7.3 Electrical inertia analysis for powerstationsandtransmissionlinesintheIEEE300bussystem. Transmissionpowercomputedfrompowerflowequationsingreenversustheinverseofinertiainblue, showingoppositerelationshipbetweentransmissionpowerandinertia. Thefollowingfactsareconcluded: (1) The grid is locally negatively curved with respect to some lines and buses. (2) Line number 1 has zeroinertiaandsolelyconnectingthenetworktotheswingbus. (3)Linesnumber0to50areinhigh centralitywith respect to the fluctuations ofpowersupply/demandinbuses. 7.5EvaluationResults 199 0 50 100 150 200 250 300 1 1.5 2 2.5 3 3.5 4 Node number in Virtual Network (Shortest Path Resistance) / (Effective Resistance) Indication of some areas in the network being Locally Negatively Curved Fig. 7.4 Effective resistance (electrical distance) curvature analysis in the IEEE300 bus system. The valueofR(a,i)/R eff (a,i)forfivesamplenodes ainvirtualresistivenetwork,whereichangesamong all other nodes. By Th. 7.3, the flat ratio is a symptom of local negative curvature for some transmission linesconnected to the correspondingnodea. 7.5 EvaluationResults Let us define the electrical betweenness centrality of a bus (resp. a line) as the inverse of its moment of inertia. We evaluated the theoretical results of the previous sections through the IEEE 300 bus case. In Fig. 7.3 compares the apparent power flow in different transmission lines together with the betweenness centrality of each line. The operation risk factor is assumed the same for all buses. It clearly shows that there are some lines with high betweenness, i.e., lowmomentofinertia,operatingunderhighpowerflow. Asshown,ourtopologymeasureis remarkablyaccurateatanticipatingtheoverloadedbusesandlines. For five specific nodes x as samples, Fig. 7.4 shows the ratio of the shortest path resistance R(x,k)totheeffectiveresistance R eff (x,k),eachofthemcalculatedbetweenthatspecificnode x and all other nodes in the virtual network. This quantity, as discussed in Th. 7.3, is a measure ofhyperbolicityforthenetwork. Twocurvesshowthat,tomostextent,theeffectiveresistance 200 NegativelyCurvedPowerGrids (electrical distance) proportionally changes with the shortest path resistance (geodesic distance). This always happens in an ideal tree, and when partly happens in a non-tree, is symptomatic of a negatively curved network. The other curves, instead, reveal more like Euclidean resistive patterns,thoughdifferentquantitative curvaturescanbeassociatedwiththem. Although,becauseofthelackofinformation,wearenotabletoanalyzethenetworkreliability inacompleteway,someconclusions canbedrawn: – Thegridislocallynegativelycurvedwithrespecttosomelinesandbuses. Thisobservation isconfirmedinFig.7.4bythesymptomsofhyperbolicityinthevirtualnetwork(Th.7.3), andinFig.7.3bythenon-uniformityofmomentofinertia(Cor.7.1). – Line number 1 has zero moment of inertia with 458 MVA transmission load. This line is theonlyoneconnectingthepowergridtoitsreferencebus,namelyswingbus. – Lines number 0 to 50 are in high centrality with respect to the fluctuations of power supply/demand at buses. To have a reliable transmission system, these lines must operate quitefarfromtheirratedcapacities. – Linesnumber51to180areinmediumcentrality,whereacollectionoflineswithhighest transmittingpoweroccurs. The transfer of these observations to Smart-Grid practice would entail the development of that technique for real grid, rather than the IEEE bus systems, and the integration of that techniqueinabiggersystemabletoreroutepowerincaseofoverloadedlines. 7.6 SmartPowerScheduling and Routing The results indicate that our preliminary research has been successful at linking curvature with line overload and at identifyinglines with heavypower flow asthose relative towhich the grid momentofinertiaislow. Havingidentifiedthecause,thequestionnowis,Whatisthecure? Theculpritinthelineoverloadingproblemisexistenceofapairofbuseswiththeireffective resistance,definedbyoneofthevirtualresistivenetworks,closetotheresistanceoftheleast resistivepath,indicativeofnegativecurvature(Th.7.3). Existenceofsuchapath—amongother pathsthathaveadifferent R eff versusshortestpathresistancebehavior—pointstononuniform curvature. Clearly, uniformizing the curvature is a step in the right direction. To be more specific,consider (7.16),fromwhichtheRiemannianmetricds 2 =Bcos(dθ )(V 2 dθ 2 +dV 2 ) emerges as an ohmic loss metric. This metric is relevant, as currents in a resistive network are distributed in such a way as to minimize Ohmic losses [132, 133, 71]. Recall that, by a theoremofRiemanniangeometry[134,Chap. 1,Lemma1.4.4],min R ds(lengthminimization) 7.6SmartPowerSchedulingandRouting 201 Generator Robustness Power Demand Power Capacities Apparent Power Power State Estimation Topological Optimization Power System Operation Power Scheduling Optimal Power Flow Demand Response Optimization Loop Power Flow Optimization Loop Fig.7.5Overallarchitectureoftopologycontrolinasmartpowergrid,displayingtherequiredinteraction withother control and operation componentsofpowersystem. andmin R ds 2 (energyminimization)havethesametrajectory. Therefore,theelectricalflow is the geodesic flow min R ds for the metricds 2 . Depending on the susceptance and voltage parametersthatappearinthemetric,thecurvaturecouldtakemanyforms. Itisagenericfact [35, 36] that, should this metric be negatively curved, the geodesic flow (current flow in the virtual grid, power flow in the real grid) will have will have some concentration points, and thatthesameconcentrationpointswillberemovedbymanipulatingthemetrictoapreferably uniformlypositivelycurvedone. 7.6.1 Ricciflow TheRicciflowisamathematicaltechniquetomanipulateaRiemannianmetricsoastouniformize the curvature of a topological surface, even a 3-manifold [135,136], subject to such topological invariantsas the Eulercharacteristic. More closelyrelated to thisproposal, a version[118] has been developed that endeavors to uniformize the curvature of a (sub)triangulation of a compact surface bymanipulationofthelinkweightw. As we saw in Ch. 6, the curvature of an interior vertexa of a triangulation is obtained from thelinkweightsw ij ,w jk ,w ki andthecosinelawdefinedas K a :=    2π − P f abc ∈F θ bc a a / ∈∂G π − P f abc ∈F θ bc a a∈∂G. 202 NegativelyCurvedPowerGrids Delta → Star Star → Delta Fig. 7.6 Star-Delta transformationstakingK 5 toaplanargraph. whereθ bc a is the interior angle of thef abc triangle at the apexa. The Ricci flow manipulates the metricw soastouniformizeK. Curvatureuniformizationhasseenitsfirstnetworkingapplicationinloadbalancing[ 35,36] andinsecuringsuccessfulgreedypacketforwardinginsensornetworks[137]. In the Ricci flow, the metric is computed as w 2 ab = r 2 a + r 2 b + 2r a r b cosϕ ab , the flow is initializedasr a (0)=1 anditeratedas dr a (t) dt =− K a (t)− ¯ K a r a (t) whereK a (t) is thecurvature of vertexa computed fromthe current metricdata, and ¯ K a is the targetcurvatureatnodea. TheaboveODEisrununtilitconvergesandthecurvatureisconverged to target curvaturesat the nodes. Load balancing [36]and successful greedy forwarding[137] weretextbookapplicationsoftheYamabe[117]andRicciflow[ 118],respectively;however,the powergridapplicationtotransferthepowerevenlyacrossthegridisnotquiteso! Thefollowing subsectionsaddresspoint-by-pointthechallengesfacingus. 7.6.2 Star-deltatransformations The grid virtual networks are certainly not planar and, even though any graph can be embedded in the compact surface of genus g (see [112]), it is highly unclear whether that can be done isometrically. So, the starting point of the Ricci flow, the triangulation of a surface, is not to be taken for granted for a power grid. The underlying electrical engineering problem, however, allows us to circumvent this difficulty. Indeed, a remarkable feature of resistive networks is that they can always be transformed to planar networks by star-delta transformations. Kuratovsky’s theorem[138,Th.11.13]assertsthatagraphisplanariffitdoesnotcontainthecompletegraph K 5 and the bipartite graphK 3,3 . It is a relatively simple exercise to show thatK 5 andK 3,3 can bemadeplanarbystar-deltatransformations(seeFig.7.6). However,itmustbeobservedthat 7.6SmartPowerSchedulingandRouting 203 suchatransformationinvolvesatleasttheeliminationofonenode. Indeed,thegenusg isan invariant; for a nonplanar graph,g> 1, while for a planar graphg = 0, so that some change inthegraphtopologyisnecessary. But,eventhoughthestar-deltatransformationchangesthe graphtopology,itdoesnotaffectthetrafficatthenodesthatarepreserved. 7.6.3 Triangulationcondition If the planar graph resulting from the star-delta transformations is a triangulation, that is, its facesaretriangles,thenuniformizingthecurvatureisatextbookapplicationoftheRicciflow [118], being cautious to properly utilize the formula on the outer boundary of the triangulation. Buttheproblemisthatitisnotguaranteedthattheplanargraphresultingfromthestar-delta transformationsontheS-virtualnetworkisatriangulation. Itmighthavefacesthatarepolygons. If the graph is rigid [139, 140], the angles can still be uniquely computed from the metric data, thecurvaturecanstillbecomputedalongthepolygonsasboundaries,andtheRicciflowcan still be run. If, on the other hand, the graph resulting from the star-delta transformations is not a Lamangraph[140],thenitisunclearhowtoproceed. 7.6.4 ConstrainedRicciflow Givena complex power supply/demandprofile S i ,thepowerflowequationsaresolved(using, say, Newton-Rapson procedure), giving the (V i ,θ i ) profile, itself providing the edge weight w ij =1/(B ij V i V j )oftheS-virtualnetwork. Sincedatasatisfyingthepowerconstraintshave already been obtained, we do not want to discard it altogether and start the Ricci flow from r i (0) = 0. Contrary to the mathematical repertoire, here, we initialize the Ricci flow so that w ij (0)=1/(B ij V i V j ),butthisrequiressolvingtheequation 1/(B ij V i V j )= q r 2 i +r 2 j +2r i r j cosϕ ij , runthealgorithm,andfactorthenewweightasw ij (t)=1/(B ij (t)V i (t)V j (t),andthenimple- mentthechangeofvoltagesandadmittancesusingFACTS. Clearly, there are two ways to manipulate these weights. Either we manipulated the line susceptance, which can be done by the Flexible AC Transmission System (FACTS), or we manipulate,totheextendpossible,thebusvoltagesVi,V j ,whichcanbedonewithsynchronous generator excitation and/or on-load tap changer and/or by Static Var Compensator (SVC) of FACTS.Acombinationofbothisalsopossible. 204 NegativelyCurvedPowerGrids Next,asweiteratetheRicciflow,therearetwoconstraintstobewatched: 1) Inthetransmissionpartofthegrid,voltagemanipulationcanbedone,butatthedistribu- tionpartofthegridthevoltageshouldremainwithinspecification( ±5%variation). 2) Thenewprofile 1/(B ij (t)V i (t)V j (t) mustsatisfythepowerconstraints. The main challenge consists in finding a strategy when a constraint is hit: Do we restrict the flow to follow the boundary of the constraint? Our ability to uniformize the curvature is limited, butit isexpected thatuniformizingthecurvaturewithinourconstraintswillmakethe transmissionmoreuniform. 7.6.5 Localversusglobal Clearly, there are questions as to whether curvature uniformization would really avoid such phenomena as the flat parts of the curves in Fig. 7.4. The deeper issue is that the Ricci flow dealswithlocalcurvature,whilesuchthingsastheflatcurvesinFig.7.4areaglobalissue. The connectionbetweenlocalandglobalcurvaturestillremainstobeclarified. 7.7 Conclusion This chapter has targeted the application of congestion analysis developed for negatively curved data networks to similar phenomena in power grids. We have investigated the geometry of manifold underneath the network as a topological analysis tool to predict the behavior of power grid in the presence of uncertain disturbances in load and generation. We have shown that line overload in the power grid can be approached with the same geometrical tools as in congestion indatanetworks. Theculpritisnegativecurvature,creatingcongestion,wherethecongestion pointsareidentifiedaspointsoflowgridinertia. Inpowergrid,“congestion”occursatlines, whereas in data networks it occurs at routers. The challenge is then to adjust the bus voltages so as to have uniform transmission load, which has to be done while remaining within the confines of voltage stability. We believe that this research, beyond its application to the power grid, canopenanewwaytoemploygeometricaltoolsinrevealinghiddenbehaviorofpowerflow networks,inwhichthetrafficcannotbequantifiedbyonlyonevariable. 7.8FutureWork 205 7.8 FutureWork 7.8.1 Developmentofarigorousmathematicaltheory The problem is that many of the concepts developed here, like “electrical inertia,” are based on classicalnetworktheory[141]intuition,butlackmathematicalformalization. WhileRiemannian geometryandits Gromovcoarse version[120]telluswhatcurvatureandinertiaareatapoint of a manifold or at a vertex of a graph [142], there are not such concepts as curvature and inertia along a curve of a 2-manifold or along an edge of a graph. (Feng Luo [117] defines a combinatorial scalar curvature along an n-simplex σ n of a simplicial decomposition of a manifold;inparticular,thisyieldsacurvaturealonganedgeoftheskeletonofamanifoldof dimension3atleast,soastohaveenoughspacetodefinedihedralangles. We,ontheotherhand, wouldneedcurvaturealonganedgeofa2-manifold. Thereliesthedifference.) Apossibility left for further research would be to define inertia/curvature along an edge via the dual network [141,Sec.42–48]. 7.8.2 Topologyimpactonstateobservabilityofpowergrid Stateestimationiscriticalforpowersystemmonitoringandcontrol. Astateestimatordetermines voltagemagnitudeandphaseateachbususingavailablemeasurementsincludingvoltage/current magnitudes/angles and power flows/injections, which are provided by conventional Remote TerminalUnits(RTU)ortheemergingPhasorMeasurementUnits(PMU). Two critical issues with a state estimator are the convergence property and estimation error. Theweightedleast-squarestateestimationmethodisbroadlyusedinpowerindustry,inwhich foragivensetofmeasurements, ˆ z =h(x)+e where ˆ z isthemeasurementvectorofsizem,xisthestatevectorofsizentobeestimated,his avectorofnonlinearfunctionsrelatingthestatestothemeasurements,andeisthevectorof measurement errors with normal distribution and of sizem. Under the assumption of additive, Gaussian measurement error, the maximum likelihood estimatex ∗ is found by solving the unconstrainedminimizationproblem minJ(x)= 1 2 ˆ z− h(x) ⊤ R − 1 ˆ z− h(x) 206 NegativelyCurvedPowerGrids whereRisthediagonalmatrixoferrorcovariances. Thenasanecessaryconditionforoptimality, anysolutionmustsatisfythefirstordercondition H(x) ⊤ R − 1 ˆ z− h(x) =0 withH beingthemeasurementJacobianmatrix. The aim is to analyze the effect of grid topology on the minimum achievable estimation error. Specifically, examinehowtheoptimalestimationerrorof anodevariablegrowswithits distance from the observed node. One also wants to examine how the structure of the power networkaffectsthestabilityofthestateestimationanditssensitivitytomeasurementerrors. 7.8.3 Topologyimpactonelectricalsecurityofpowergrid Large-scale blackout caused by cascading failure is an intrinsic drawback of electric power transmissiongrids. Cascadingfailureisverychallengingtoanalyzebecauseofthehugenumber ofpossiblerareandunanticipatedfailuresandthedependenceofthefailuresontheprevious failuresinthesequenceoffailures. There are two different approaches to the large blackouts caused by cascading failure. One istotakeintoaccountalltheelectricalprocessesintegratedoverthewholegridandexamine the detailed sequence of failures of a particular blackout after happening. Although the analysis providesusefulengineeringdataforstrengtheningweakerpartsofthesystem,itneglectsthe behavior of electric power grid as a whole and overemphasizes the details. Another approach is toinvestigatethenetworkrepresentationofpowergridfromatopologicalperspectivewithout considering the details of special electrical processes. In this approach, the grid model is highly simplifiedandtheimportantfactorsinvolvedinthefailurespropagationarenotconsidered,such asthelineoverloadsandredistributionofpowerflow. Onemayuse,instead,virtualnetworks introduced in Sec. 5.3 which convey main electrical characteristics of thegrid, and at the same timecanbeanalyzedfromapuretopologicalperspective. Therearemanywaysinwhichfailurescaninfluencefurtherfailures,includingoverloads, hidden failures of protection systems, oscillatory instability, etc. Considering the cascading failures caused by lines overload, the conventional way of improving the security is to increase the capacity of the transmission lines. But we believe that the topology intrinsically contributes tothecollapseprobability,whichcannotbeovercomebyincreasinglinecapacities. Specifically, one may try to quantify how much close we can get to a “perfect” network if we do not change thetopology,whereaperfectnetworkcanresistallormostcontingencies. 7.8FutureWork 207 7.8.4 Powerflowcontrolversusvoltagestability ActivepowerflowcanbecontrolledusingFACTSvoltageactuators,butthisvoltagemanipulation haseffectonreactivepower,andhenceonvoltagestability. Theriskofvoltageinstabilityisdue tothenonlineareffectsintheloads. Considerableresearchhasbeendevotedtotheso-called “static-dynamic”gapinthemodelingoftheloads,toHopfbifurcationinasimpleonegenerator, oneline,oneloadarchitecture,andtosensitivityanalysisbyphasormeasurementsinthesystem. Hereweproposetoshiftthefocustowardsthecomplexfeedbacknetworkaspectsbylooking attheeffectofmanyaggregatedloadsinthediagonallyperturbedfeedback,astheexpenseof simplifying the load model to make the large-scale feedback and robustness aspects analytically tractable. Wefeelthatanequivalent linearization(describingfunction)load modelcanbethe rightcompromise. Obviously,thedescribingfunctionmodeloftheloadatbuskis(P(V k ,ω)+jQ(V k ,ω))/I k 2 , whereP(V k ,ω),Q(V k ,ω) is the experimental data. For a generic load, we haveP =K P V k a ω b and Q = K Q V k c ω d . The problem is to eliminate V k using P k 2 + Q k 2 = V k 2 I k 2 , but the onlyrigorousapproachtodealwiththenon-integernatureoftheexponentsistocomputethe Puisseuxexpansionofthedescribingfunction,fromwhichhopefullythepotentialforvoltage instabilitywillcropup. Amoresensibleapproach,however,might betoextractthe uncertain characteristics V k a and V k c in an uncertain feedback loop and attempt to put the nonlinear distortionsbetweenbounds,sothatthemulti-loopcircle/Popovcriterion,oreventheZames- Falb multiplier machinery, could be put into gear. Topology errors would be lumped in the transmissionblock. Evenwithtopologyerrorsandgeneratorexcitationcontrolloop,theanalysis oftheresultinguncertainmulti-loopisstillclassical,asfarascontrolisconcerned. Stillwithin the realmof classicalcontrol is modelreduction techniqueson thissystem that couldrevealtheslowcoherencyclustering. A bit less classical would be the understandingof the same slow-coherency techniques under fluctuations. But the definite nonclassical part is the factthattheRicci/Yamabealgorithm,whichdistributesthefluctuatingpowerthroughoutthe transmission,“tunes”thetransmissiondatainanadaptivemanner. Inaddition,thecurvature datafromthe voltage stabilityblocktotheRicci blockclosesanotherloop. Theultimate issue isprovidedbystabilityofthismultiloopadaptivesystem. Chapter8 Multi-AgentCooperativeVoltageControl Weconsidertheproblemofvoltagestabilitycontrolandreactivepowerschedulinginamulti- areapowersystem(possibly)controlledbyindependenttransmissionsystemoperators. Amajor issue in this problem is to coordinate, with a high level of robustness, the control actions of the interconnected areas with respect to their operational objectives and constraints. This work proposes an agent-based cooperative approach to this issue, where each area is treated as an intelligent agent [143] that pursues small-scale, self-owned objectives. We assume that each control area is capable to properly communicate with its own entities and to stably govern them inacentralizedordecentralizedmannerwithenoughsecurityandflexibility. Itisshownthat while each agent pursues its local goals and interests, the proposed multi-agent control scheme stabilizes the large-scale system and achieves the global goals of an adaptive voltage control functionthroughneighborhoodactiveinteractions. 8.1 LargeScaleVoltage Stability Most literature on multi-area voltage stability proposed centralized control strategies, assuming that each transmission system operators (TSO) transfers its prerogatives to a central entity whichisinchargeofbuildingconsensusamongdifferentareasthroughaspecificmulti-party optimization scheme. Looking at the opposite side, some research papers introduced purely decentralizedschemeswithnoinformationexchangebetweenTSOs,whereeachcontrolarea assumes an external network equivalent in place of its neighbor areas and optimizes its own objectivefunctionregardlessofitsimpactontheothers. 210 Multi-AgentCooperativeVoltageControl Webelievethatintheemergingpowergridwitheverlargerareaofinterconnectedoperators, thecentralizedformulationisneitherfeasible in computation nor reliablein communication. Purely decentralized approaches, on the other hand, do not lead to an optimal performance in large and can not guarantee a secure operation, where conflicting local strategies result in a reduction of TSOs’ performance criteria [144]. The idea of this work is more aligned with anotherpathofresearchthatdescribesthebenefitsofinter-TSOcoordinationforsteady-state voltage control. Specifically, active-interaction-based distributed control approaches are highly promising in the light of access to wide-area synchronized PMUs and resilient high-speed communicationnetworksinthefuturesmartgrid. 8.2 Contribution Akeydistinctionbetweenourmulti-agentdesignandaconventionaldecentralizedschemeis negotiation, meaning that intelligent agents do not respond to predefined requests from specific agents, but they negotiate and interact with each other in a cooperative manner to reach a “fair” agreementthatissatisfactoryforallofthem. Thisopensthedoortoadaptation,optimization, reconfiguration, and fault tolerance. However, despite these advantages, negotiation introduces anextradegreeofuncertaintyintothesystem,whichisduetogeneraldifficultyofpredictingthe futurestateofanagenttoguaranteeareal-timeperformance. Tosettlethisdifficultyweutilizea distributedmodelpredictivecontrolschemeinwhicheachagentknowsalocalmodelofitsown area as well as reduced-order quasi-steady-state (QSS) approximations of its neighbor areas. Everytime-slot, 1 eachagenttakesdecisionsbysolving alocaloptimizationproblemusingits local measurements and the latest model information received from the neighbor agents. A disposable part of local control information is then communicated to the neighbor agents to be takenintoaccountintheirnextoptimizationiteration. 1 The coordination sampling time, called time-slot, is the step-time that TSOs exchange information with their neighbors. Itisneitherinvariantnorthesamebetweendifferentagents,thoughinthisworkwetakeitequaland constantamongallagents. Further,thetimeslotshouldbediscriminatedfromthecontrolsamplingtimek used in discretizing continuous-time dynamics, which is private to each agent. The latter is much smaller than the time-slotandhastobechosensuchthatthediscrete-timeapproximationsadequatelyreflectthedynamicsofthe continuous-timemodel. 8.3Methodology 211 Fig. 8.1 ArchitectureofaTSOvoltagecontrolagent. 8.3 Methodology We represent a large-scale multi-area power system with a connected undirected graph, for which TSO control agents are the nodes and transmission lines (and thus communication links) amongthemaretheedges. 8.3.1 AgentArchitecture ATSOvoltagecontrolagentiscomposedofsevencomponentsshownbyFig.8.1. The“commu- nication” module enables the agent to negotiate with other agents for the coordinated execution of proper tasks. The “sensors” module perceives local data and estimates the voltage level and reactivepowergenerationwithintheTSO.The“actuators”moduleexecutesthetasksbysending commands to tap positions of load transformers, generator voltage controllers, FACTS devices, shuntcapacitorbanks,and/orloadsheddingprocedure. The “decision making” module evaluates the current operating state using endogenous data from the sensors and exogenous data from neighbor areas. Based on the operating state, the “control strategies” moduleprovides the “decision making” module with a proper control and optimizationalgorithmfromitsdata-base. As long as the control strategy has not changed, the “decision making” module dynamically adjusts its behavior in accordance with the information provided by the “sensors” and “com- munication” modules. Meanwhile, the “adaptation” module continuously evaluates the control policyperformanceandaccordinglyupdatesthemodelparametersandobjectivefunctionsin the”controlstrategies”data-base. 212 Multi-AgentCooperativeVoltageControl To meet voltage control requirements in different system states, the control agent should take different control strategies and algorithms under different conditions. In this work we only considertwostatemodes,i.e.,thenormalmodeandtheemergencymode. 2 8.3.2 EmergencyModeOperation When system runs into emergency resulting from a large disturbance, it is necessary to manage a fast, dynamic response for providing the bus where voltage violation occurs with reactive power support. In this mode, the control agent makes its own decision to change the settings of reactivepowerinjectionthrougheveryindividualgenerator,synchronouscondenser,andfast responding static VAr compensator to rapidly restore the abnormal voltage back to its allowable range. If voltage violation cannot be removed by the agent itself, it sends request for voltage controlassistancetoneighborareas. For the coordination of neighboring TSO agents in recovering the violated voltage, we employ the contract net protocol (CNP) which is widely used in multi-agent systems. It is a negotiation-basedprotocolto establishefficientcooperationamongagents,verysimilartothe broadcasting protocol in communication networks. If one agent discovers a problem that is not abletosolveitalone, itannouncesthisproblemtotherestoftheagents. Someotheragentswill reply to the corresponding agent to provide help for solving the problem through a bidding and contractapplication. 8.3.3 NormalModeOperation Inasteady-statepractice,eachTSOischaracterizedbycomplexinteractionsbetweencontinuous dynamics and discrete events. We assume that each TSO uses a general dynamic model of its own area as well as a reduced-order QSS model of its neighbors. The local area model is representedby aset ofdifferential-algebraicequations (DAEs)consistof short-termdynamics (due to synchronous generators and their regulators, induction motors, and FACTS devices) and long-termdynamics(duetophenomena,controllers,andprotectingdevicesthatacttypically over several minutes following a disturbance), constrained to network power flow algebraic equations[145]. Ateachtime-slottheneighborTSOsexchangereduced-orderlinearizedmodels that reflect QSS approximation of their long-term dynamics. Specifically, these reduced-order 2 In a more comprehensive model, each agent is equipped with a transition function module that given the systemstate,decisions,andexogenousinformation,determinesthestateatthenexttime-slotusingasetofequations and/orrulesthatdescribehowthesystemislikely to evolve over time. 8.3Methodology 213 models prevent disclosing confidential local information such as economical cost functions and portfoliocondition. Every control time-stepk, the continuous-time linearization of local DAE equations around the operating point is obtained in the “decision making” module, where required Jacobians caneither be derived analytically orcomputed numerically. Weassume smallvariationsof the variablesaroundwhichthemodelislinearized. Otherwise,modechangeshavetobeconsidered or a new dynamic model has to be called from the “control strategies” module. Then the continuous-time linearized model is discretized to yield the following discrete-time control modelintheaffineexpressionsofhyper-vectors x(k) andu(k) as x i (k+1)=A ii x i (k)+ X j∼ i A ij x j (k)+B i u i (k)+g i v i (k)=C ii x i (k)+ X j∼ i D ij u j (k− 1)+h i (8.1) where the vectorsx i ,u i andv i respectively represent local continuous state variables, local continuouscontrolinputs,andlocalalgebraicvariablesoftheithagent,k isthediscretetime step,andthesymbol∼ denotesadjacency. The obtained discrete-time approximation is employed as a prediction model in the optimal control problem formulation, subject to appropriate hard constraints on the inputs. Every time- stepk,thecontrolalgorithmsolvesanoptimizationproblemofthefollowingformtofindthe controlaction: min X i (k),U i (k) J i X i (k),U i (k) s.t. 1) equalityconstraints(8.1) 2) inequalityconstraintsontheinputs whereX i (k)={x i (k+1|k), ..., x i (k+l|k)}andU i (k)={u i (k|k), ..., u i (k+l− 1|k)}. Thisperformanceindexrepresentsthemeasureofthedifferencebetweenthepredictedbehavior andthedesiredfuturebehavior: thelowerthevalue,thebettertheperformance. Thevariables x i (k+m|k) andu i (k+m|k) arerespectivelythepredictedstateand thepredictedcontrol of agenti at the time-stepk+m given the information at the stepk. The optimization scheme producesanopen-loopoptimalcontrolsequenceinwhichthefirstcontrolvalueisappliedto the system, i.e.,u i (k)=u i (k|k). Then the controller waits for the next time-step to repeat this process,findingthenextcontrolaction[146]. 214 Multi-AgentCooperativeVoltageControl Fig.8.2NormalModeOperationbasedonDistributed ModelPredictiveControlapproach. Inasteady- statepractice,eachTSOusesageneraldynamicmodelofitsownareaaswellasareduced-orderQSS modelofits neighbors, exchanged at eachtime-slot. Everytime-step,eachTSOagentbroadcastsonlyapartofitssolutiontothelocaloptimization problemafterapplyingitscontrolactionsinitslocalarea. Inthecomputation,agentsusethe information they get from neighbor agents to estimate the effect from neighbor subsystems, meaning that each agent uses the predictions of neighbor agents at the previous time-step to estimate the influence of neighbor TSOs. It is shown that this control and optimization strategy guarantees large scale stability and fairness, while also satisfies the local objectives. 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Abstract (if available)

Abstract Complex networked systems appear in almost every aspect of science and technology. Most social, biological, informational and physical networks are large assemblies of nonlinear dynamical systems interacting via non-trivial topologies. Many of these networks are complex in different ways including that they are large in scale and so hard to visualize

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Creator Banirazi, Reza (author)

Core Title Control and optimization of complex networked systems: wireless communication and power grids

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/18/2018

Defense Date 08/30/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag complex networks,OAI-PMH Harvest,packet routing,power grids,processing networks,queue stability,stochastic network optimization,wireless networks

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jonckheere, Edmond (committee chair), Bonahon, Francis (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email banirazi@gmail.com,banirazi@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-84268

Unique identifier UC11675417

Identifier etd-BaniraziRe-6879.pdf (filename),usctheses-c89-84268 (legacy record id)

Legacy Identifier etd-BaniraziRe-6879.pdf

Dmrecord 84268

Document Type Dissertation

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Rights Banirazi, Reza

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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