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University of Southern California Dissertations and Theses
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Hyperbolic geometry of networks
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Hyperbolic geometry of networks
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HYPERBOLIC GEOMETRY OF NETWORKS Copyright 2003 by Poonsuk Lohsoonthorn A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2003 Poonsuk Lohsoonthorn R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UMI Number: 3116747 Copyright 2003 by Lohsoonthorn, Poonsuk All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3116747 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-1695 This dissertation, written by Poonsuk L o h so o n th o rn i s under the direction o f h dissertation committee, and approved by all its members, has been presented to and accepted by the Director o f Graduate and Professional Programs, in partial fulfillment o f the requirements fo r the degree o f DOCTOR OF PHILOSOPHY Director Date A u g u st 1 2, 2001 Dissertation Committee Chair R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. DEDICATION Dedicated to my parents, Paiboon and Somchit, for their love and support through out all phases of my life. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. ACKNOWLEDGMENTS I would like to express my sincere gratitude to my academic advisor, Professor Edmond A. Jonckheere, for his encouragement and supporting throughout this project. This task could not have been completed without his generous advice. I would like to thank the members of my dissertation committee, Professor Fran cis Bonahon, for his knowledge of differential geometry and topology, and Professor Bhaskar Krishnamachari, for his kindness and useful comments. I would also like to thank the members of my qualifying committee, Professor Jean-Luc Gaudiot, Pro fessor Joao P. Hespanha, and Professor Wayne Raskind for their time and useful comments. Finally, I would like to thank my parents, Paiboon and Somchit for their endless love, encouragement, and unconditional support. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. iv TABLE OF CONTENTS D e d ic a tio n .................................. ii Acknowledgments .................................................... iii List of F i g u r e s .............................................................. vi List of T a b le s ............... x A b s tr a c t ................................................................... xi Chapter 1 Introduction 1 Chapter 2 Embedding a graph on a surface 7 2.1 Representation of graphs ................................. 7 2.2 Topological surface ................................. 14 2.3 Embedding of a graph into a surface ........................... 19 2.4 Algorithm for minimum em bedding................................................. 22 2.5 Cayley g r a p h ........................ 27 2.6 Weighted directed g r a p h .............................. 30 Chapter 3 Riemannian Geometry 33 3.1 Differential manifolds .................... 33 3.2 Riemannian manifolds ................................. 37 3.3 Riemannian Connections . ................................. 40 3.4 G eodesics................. 42 3.5 Curvature ........................... 49 3.6 Jacobi fields . ........................... 53 3.7 Hyperbolic spaces with constant curvature ... . .. .. .. .. . 55 Chapter 4 Coarse Geometry 60 4.1 Geodesic metric sp aces..................... 60 4.2 Comparison spaces ........................ 63 4.3 J-Hyperbolic spaces ........................ 71 4.4 Quasi-Geodesics in Hyperbolic space ........................... 78 4.4.1 Riemannian geometry c a se ............... 82 4.4.2 Geodesic metric space case ............... 87 4.5 fc-local Geodesics in Hyperbolic space .............................. 88 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. V Chapter 5 Computation of metric curvature in Riemannian Ge ometry 90 5.1 Hyperbolic trigonometry ........................................... 91 5.2 Slimness computation .............................................. 95 5.3 Insize computation ........................ 98 5.4 Thinness computation ..................... 102 5.5 Fatness computation ........................... 104 5.6 Billiard dynamics interpretation of fatness of hyperbolic geodesic triangles ........................................ 105 5.6.1 Euclidean triangular billiard table ................................. 106 5.6.2 First order conditions..................... 108 5.6.3 Hyperbolic orthocenter construction . ............................... 113 5.6.4 Uniqueness .................................. 117 5.6.5 Second order variation ..................... 123 5.6.6 Fatness form ula........................... 131 5.6.7 Obtuse angle case.................. 134 Chapter 6 Random graph and hyperbolic graph 137 6.1 Random graphs modeled by Erdos and Renyi ............... 137 6.2 Small worlds graph modeled by Watts-Strogatz . . . . . . . . . . 144 6.2.1 a-model ................................................. 145 6.2.2 /3-model ......................... 146 6.3 Barabdsi-Albert scale free network.................................................... 148 6.4 Simulation .............................................................................. 151 6.4.1 Random graph as modeled by Erdbs and R e n y i................ 155 6.4.2 Small worlds as modeled by W atts-S trogatz...................... 160 6.4.3 Barabdsi-Albert scale free network .............................. 164 6.4.4 Growth with uniform attachment graph . . . . . . . . . . 168 6.4.5 Comparison .................. 169 Chapter 7 Application of hyperbolic geom etry to communication networks 176 7.1 Curvature in communication networks ........................................... 176 7.2 Metric graph and Geodesic space ........................ 182 7.3 Routing algorithm via fc-local g eo d esic............... 184 Chapter 8 Conclusion 187 8.1 Research summary ........................................... 187 8.2 Future Perspectives .................................. 188 Bibliography 191 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. vi LIST OF FIGURES 2.1 Variations of Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Polygon representations of sphere, torus and projective plane. . . 16 2.3 Embedding graph into a sphere by pulling handle........................... 19 2.4 The nonplanar graph K$ and ..................................................... 21 3.1 Image of v = t (cos 9, sin 9), 0 < t < 1, 9 = 0, ..., under the exponential map at 0.5i for the upper half plane. ...................... 58 3.2 Image of v — t (cos 9, sin 9), 0 < t< 1, 9 — 0, ..., under the exponential map at 0.5 (1 + i) for the Poincare unit disk. . . . . . 59 4.1 A geodesic triangle with the corresponding angles. ................. 67 4.2 A ds-slim geodesic triangle ............................................................. 72 4.3 A tree is ($s = 0)-slim. The right hand triangle is (hs = l)-slim (assuming that the length of every link is 1)......................... 72 4.4 The internal points of A and the tripod Ta .................. 74 4.5 Fatness of geodesic triangle ............................................................. 75 4.6 Envelope of all distance plots between (A = 2, e = 0)-quasi-geodesics and a geodesic of length 20. The curvature k = — 0.05 is adjusted so that the solution reaches the bound, resulting in a continuously differentiable curve................................................................................. 84 4.7 Envelope of all distance plots between (A = 2, e = 0)-quasi-geodesics and a geodesic of length 20. The curvature k = — 0.005 is adjusted so that the solution does not reach the bound, resulting in a non- differentiable curve................................................................................. 84 4.8 Comparison of Hausdorff distance upper bound: Gromov’s tradi tional bound, refined bound in Geodesic metric space, and refined bound in Riemannian manifold............................................................ 88 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. vii 5.1 The Fagnano period orbit, shown to correspond to the orthic triangle. 106 5.2 The Pejer construction, showing that the orthic triangle is the pe riodic three orbit. The construction remains valid in hyperbolic geometry. .............................................. 108 5.3 The Fejer construction for the obtuse angle case. The construction remains valid in hyperbolic geom etry................................... 134 6.1 The backbone graphs in the simulation.......................................... .... 153 6.2 The degree probability distribution function that results from the numerical simulation of random graphs with different backbones. . 156 6.3 The degree probability distribution function that results from the numerical simulation of random graphs with Star backbone. . . . 156 6.4 The degree probability density function that results from the nu merical simulation of random graphs with different backbones. . . 157 6.5 The degree probability density function that results from the nu merical simulation of random graphs with Star backbone 157 6.6 Comparison of E (Sq) for random graphs of order 50........................ 158 6.7 Comparison of E for random graphs of order 50. . . . . . . 159 6.8 Comparison of E (So) for random graphs of order 100. . . . . . . 159 6.9 Comparison of E (-J ^ ) for random graphs of order 100.. 160 6.10 The degree probability distribution function that results from the numerical simulation of small world graphs with different parameter P- ........................................................ 161 6.11 The degree probability density function that results from the nu merical simulation of small world graphs with different parameter P........................ 161 6.12 Comparison of E (Sq) for small world graphs of order 50................. 162 6.13 Comparison of E for random graphs of order 50. ............. 163 6.14 Comparison of E (So) for small world graphs of order 100 . 163 6.15 Comparison of E (^ f^ ) for random graphs of order 100. . . . . . 164 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. viii 6.16 The degree probability distribution function that results from the numerical simulation of scale free graphs with different backbones. 165 6.17 The degree probability density function that results from the nu merical simulation of scale free graphs with different backbones. . 165 6.18 Comparison of E (8q) for scale free graphs of order 50. ........... 166 6.19 Comparison of E for scale free graphs of order 50..... 166 6.20 Comparison of E (5q) for scale free graphs of order 100..... 167 6.21 Comparison of E for scale free graphs of order 100... 167 6.22 The degree probability distribution function that results from the numerical simulation of growth with uniform attachment graphs with different backbones..................................... 168 6.23 The degree probability density function that results from the nu merical simulation of growth with uniform attachment graphs with different backbones....................................... 169 6.24 Comparison of E (Sq) for growth with uniform attachment graphs of order 50....................................... 170 6.25 Comparison of E for growth with uniform attachment graphs of order 50. ..................................................... 170 6.26 Comparison of E ( 8q ) for growth with uniform attachment graphs of order 100. ........................................................ 171 6.27 Comparison of E for growth with uniform attachment graphs of order 100.......................................................................... 171 6.28 Comparison of E (8q) for all random graph generators of order 50. 172 6.29 Comparison of E for all random graph generators of order 50. 173 6.30 Comparison of E {So) for all random graph generators of order 100. 173 6.31 Comparison of E for all random graph generators of order 100. .................................. 174 7.1 The topology of the Internet at router level. .... .. .. .. .. 178 7.2 The topology of the WWW: @mathlab.usc.edu. . . . . . . . . . . 178 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. ix 7.3 The ISP graph consisting of a highly connective core and long ten drils. Observe that the geodesic triangle, A A B C is slim. . . . . . 181 7.4 Construction fc-local geodesic via recursively concatenating Mocal geodesic (in this case k = 2). 185 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. X LIST OF TABLES 2.1 Euler characteristic of various compact su rfa c e s..................... 18 2.2 Genus and maximum genus of some families graphs . . . . . . . . 27 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. ABSTRACT The concept of curvature of communication networks is investigated through the theory of <5-hyperbolic space, which can be intuitively viewed as the generalization of Riemannian manifolds with negative curvature to metric graphs. The hyperbolic measure 5 can be expressed in terms of slimness, insize, thinness, and fatness of geodesic triangles in the metric space. The analytical formula for the slimness, insize, thinness, and fatness are computed in terms of the internal angles of the geodesic triangles and the curvature k of the underlying Riemannian manifold with constant negative curvature k. In addition, the fatness of a geodesic triangle with acute angles only can be given a billiard dynamics interpretation, in the sense that the optimum inscribed triangle is the period three orbit of the billiard dynamics on a geodesic triangular table. To assess the hyperbolic property of communication networks, the mathematical expectation of 8 over the diameter for several random graph generators is computed by Monte Carlo simulation. Among random graphs, small world graphs, and scale free generators, the scale free model, which is used as a topology generator in communication network, appears to be the most hyperbolic. This result is an extra R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. piece of evidence of the hyperbolic property of the internet which has already been claimed by two different groups, using other arguments, though. With the evidence of the ^-hyperbolic property of the internet, multi-path rout ing can be achieved along quasi-geodesics, which can be computed via Mocal geodesic paths. It turns out that the alternative paths are sufficiently close to the optimum path. To assess the closeness between geodesic and quasi-geodesics, an upper bound on the Hausdorff distance between the geodesic and quasi-geodesics is derived for Riemannian manifolds with constant negative curvature and general 5-hyperbolic geodesic spaces. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 1 Chapter 1 Introduction The objective of this thesis is to develop a strong mathematical foundation for communication networks focusing on hyperbolic geometry. The Internet and World Wide Web (WWW) are probably the most inspirational communication networks of this century. The internet can be considered as a graph where the vertices are cor responding to computers and routers, and the edges are corresponding to the wires and cables that physically connect vertices. Similarly, the WWW can be considered as a graph where the vertices are corresponding to web documents, and the edges are corresponding to the directed hyperlinks (URL’s). Due to the substantial size and complexity of the Internet and WWW, just about all the research on network topology has mainly been focusing on the field of random graphs. The self-similar behavior of the Internet traffic signals ([LTWW94] and [PF95]), a manifestation of complex behavior outside the realm of Gaussian processes, has given a boost to the theory of nongaussian a-stable distributions (see [ST94, Chap. 7 ] for self similar processes). Although the random aspects of the internet grid are reasonably well understood, it is argued that this is not quite so for the geometric topology R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 of the network graph. The geometric topology of graphs refers to a new field of mathematics that lies within the intersection of the theory of embedding graphs in such low dimensional continuous geometric objects such as surfaces, and the large scale behavior of graphs encapsulated in the concept of Gromov-hyperbolic graphs [Gro87]. The embedding of the internet graph into continuous geometric objects intu itively follows from an attempt to go around the overwhelming curse of dimension ality in the internet. It is reasonable to approximate the internet grid with some sort of continuous geometric structure such as a surface or a manifold. Without considering the metric structure, the embedding of graphs on surfaces is a fun damental problem in graph theory, which mainly focuses on the topological and algebraic aspects of a graph. However, if a graph is embeddable in a surface of genus greater than 2, then this surface can be endowed with a hyperbolic metric and the graph written on the surface has good chances to be hyperbolic from the large scale. Although embedding graphs on surfaces is a traditional problem, the emergence of coarse geometry as developed by Gromov [Gro99] and noncommutative geometry as developed by Connes [Con94] have given a much broader interpretation to graph surface embedding, in which a surface is viewed as resulting from observing the graph through varyingly blurring lenses. Network graphs acquire their unique significance through the traffic that they R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 carry and which follows cost minimizing paths. Reinterpreting the traffic in the coarse geometry context in which graphs and surfaces are coarsely equivalent im mediately leads to the mathematical conceptualization of traffic as a geodesic flow in the dynamic meaning of the term. That is, the link flow can be extended to a vector field over the tangent bundle and the information flow can be considered as a hydrodynamic flow on a manifold, which could possibly be described by partial differential equations (PDEs) on a surface or a manifold. With the development of the PDE theory by Gromov [Gro86], the solutions of PDEs are rather dense in the space of functions and can get almost everything, even fractals. This PDE theory is probably suitable to explain the self-similar behavior of the Internet traffic signals. In Riemannian geometry, a geodesic is a curve that joins two points and that has its tangent vector parallel to itself. It can be shown that a geodesic in a Rie mannian manifold is locally arc length minimizing. In network graphs, a geodesic is a minimum communication cost path from a source vertex to a destination vertex. These two concepts-geodesics on graphs and manifolds-can be unified in coarse geometry by defining a geodesic to be an isometric embedding of an interval of the real line into a geodesic metric space, which is a space endowed with a distance function such that every two points can be joined by an arc of length equal to the distance between the end points. In Riemannian geometry, the so-called curvature determines the behavior of the geodesics. In negatively curved or hyperbolic spaces, the geodesics are well behaved and not too sensitive to change of the end points, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 as revealed by the Jacobi field along the nominal geodesic. A concept in the same spirit is that of quasi-geodesics that are nearly optimal paths (up to some additive tolerance £ > 0 and some multiplicative tolerance A > 1) with the same initial and final points. To apply the concept of curvature to network graphs, the curvature must be defined by relying on the distance only, since the inner product is not defined on a network graph. It turns out that curvature in coarse geometry can be redefined in terms of the distance only as the slimness 5s of the geodesic triangles. A geodesic metric space is hyperbolic if the slimness of all geodesic triangles is uni formly bounded. A hyperbolic space in coarse geometry can be intuitively viewed as a negative curvature space in Riemannian geometry and the slimness of geodesic triangles is the fundamental metric that encapsulates the hyperbolic, i.e., the neg ative curved, property. The smaller the slimness, the more negatively curved the geodesic metric space is. Among the attributes of a hyperbolic space is that the quasi-geodesics are guaranteed to remain close to geodesics by a bounded constant R (5, (A , e)) [BH99]. In addition, a fc-local geodesic, which is a locally optimal path such that the subpath that connect every two points of distance equal to or less than k is in fact the geodesic, is a quasi-geodesic where the tolerances (A, e) depend upon 5 and k. A question is, How likely is it for a network graph to be hyperbolic? Network graphs are known to have the power law degree distribution [FFF99] which can be generated from the growth with preferential attachment process as in the Barabdsi- R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 5 Albert scale free model [BA99]. To assess the hyperbolic property, the expected delta over the diameter, E , of several random graph generators were com puted in Monte Carlo simulation. It appeared in the simulation that the scale free model is likely to be more hyperbolic than any other random graph generators. This result demonstrates the hyperbolic property of network graphs as in scale free generator. The most promising potential of hyperbolic geometry in terms of network secu rity is the formulation of multi-path routing so as to "harden" the network against eavesdropping and packet sniffing. However, the Transmission Control Protocol (TCP) is not very robust against out-of-order packet arrival. It is hence imperative to send the packets along nearly optimal routes having delays about as small as the delay of the optimal route. Indeed, those nearly optimal paths are formalized as quasi-geodesics and can be computed as fc-local geodesics. This thesis is organized as follows. In Chapter 2, the fundamental concepts of graph theory are introduced, mainly focusing on the traditional problem of em bedding a graph on a surface. In addition, this chapter introduces the Cayley graph which provides a prototype of hyperbolic graphs. In Chapter 3, the funda mental concepts of Riemannian geometry are discussed, including the concept of geodesics, curvature, and Jacobi field. In Chapter 4, the coarse and J-hyperbolic spaces are introduced for geodesic metric spaces. In particular, the concept of curvature is defined for geodesic metric spaces without the restriction of Rieman- R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. man manifold structure. The <5-hyperbolic spaces are, intuitively, an extension of the concept of Riemannian manifolds of negative curvature. Then the connections among geodesics, quasi-geodesics and fc-local geodesics in <5-hyperbolic spaced are discussed. In Chapter 5, the explicit formula for the various < 5 ’s are computed for each geodesic triangle in the Riemannian manifold with constant negative cur vature k. In Chapter 6, the connections between 5-hyperbolic metric spaces and several random graph models are discussed. In particular, the relationship between d-hyperbolic spaces and scale free graphs is investigated and presented as an extra piece of evidence that the world network is negatively curved. In Chapter 7, the ap plication of hyperbolic geometry to multipath routing in communication networks is discussed. This involves the construction of fc-local geodesics as the alternate paths that provide the near optimal paths in hyperbolic spaces. Finally, the last chapter provides the research summary and a number of open problems for future work. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 7 Chapter 2 Embedding a graph on a surface In this chapter, fundamental concepts and preliminary results in graph theory are introduced, both related to the concept of embedding a graph on a surface. This chapter involves the following: First, the definition of graphs; second, the concept of topological surfaces and the classification theorem for compact surfaces; third, the concept of embedding a graph in a surface (this concept relies on the rotation system); fourth, the algorithm for minimum embedding; next, the Cayley graph, a graph that represents a group (the Cayley graph provides the connection between graph theory and group theory); finally, the weighted directed graph. The material introduced in this chapter is for the sole purpose of reviewing the concept of embedding a graph on a surface; for a considerably more comprehensive exposition of graph theory, see [GT01], [MT01], and [WhiOl]. 2,1 Representation of graphs In this section, the basic terminology form graph theory is introduced. A graph is intuitively viewed as a collection of vertices (or nodes) and a collection of edges (or R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. links) that join the vertices. The formal definition of a graph is as follows: D efinition 1 A graph G consists of a non-empty set VG of vertices and a set Eq of edges such that 1. An endpoint set VG (e) of an edge e contains two distinct vertices. 2. If e ^ e, then VG (e) ^ VG (e). From the definition of a graph, there is a 1-1 correspondence between each edge e & Eg and the endpoint set Vq (e); therefore, each edge e is sometimes denoted by {vi,v2} = VG{e). Definition 2 The order of G is \VG\ , and the size of G is \EG\ . A graph G is finite, if |Vg| < oo and \EG\ < oo. A graph is infinite if it is not finite. With few exceptions, the graphs discussed in this chapter are finite graphs. Con sequently, graphs can also be viewed as finite one-dimensional simplicial complexes. Several extensions of the concept of graphs (e.g. loop, multiple edges and di rected edge) are defined as follows: D efinition 3 1. An edge e with VG (e) containing one vertex is a loop. A loop graph allows for loops. 2. An edge e is multiple if there exists e ^ e, such that e, e € EG with VG (e) = VG (e). A multigraph allows for multiple edges. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. loop multiple edge directed edge directed graph Figure 2.1: Variations of Graphs. 3. A direction for an edge e is a function a from a set {BEGIN, END} onto Vg (e). A directed edge ea is an edge e with a direction o . A directed graph (digraph) is a graph where every edge is a directed edge. 4- A pseudograph allows for loops and multiples edges. 5. An underlying graph is a corresponding graph where all edge directions are deleted and all multiple edges are coalesced. Examples of loop, multiple edge, directed edge, and directed pseudograph is shown in Figure 2.1. A directed edge eacan be denoted by (vi,v2) where e = {vi,v2} , a (BEGIN) — vi,o (END) = v2 ■ D efinition 4 1. The vertices u and v are adjacent, if there exists an edge e € E q with Vq (e) = {u, v } . In addition, the vertex u and the edge e are incident upon each other. The set I q = {Vg (e) | e € Eq] is the incidence structure. 2. The degree (or valence) deg (v) of the vertex v is the number of edges (with each loop being counted twice) incident upon v, that is deg (v) = \{e e E g | v E VG (e)}|. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 10 A vertex of degree 0 is an isolated vertex. A graph G where each vertex has the same degree k is a k-regular graph; a 3-regular graph is called a cubic. For every graph, the sum of the degrees of the vertices is equal to twice the number of edges because each edge contributes exactly 2 to the sum of the degrees. A graph can be represented by its incident structure. In addition, a graph can also be represented by an incidence or adjacency matrix. D efinition 5 1. The incidence matrix To for a graph G is of dimension \Vq\ x \Eq\ where the Tq (i,j) entry in row i and column j is defined as follows; f 0, if V i £ V (ej) Tg (bj) — 1, if Vi £ V (ef) and \V (e^) j = 1 2, if Vi E V (ej) and \V (ej)| = 2 2. The adjacency matrix Aq for a graph G is of dimension \Vq\ x |Vg| where the A c {i,j) entry in row i and column j is the multiplicity of the adjacency between the vertices Vi and Vj. Representation of a graph by an incident or adjacency matrix may involve a loss of space efficiency compared with incident structure because of the additional zeros. However, the advantage of this representation by matrix form is the information recovery. D efinition 6 A graph H is a subgraph of a graph G if Vr CVq and E r C Eq- In addition, if Vr = Vg, the graph H is a spanning subgraph of a graph G. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 11 An important example of subgraph is a walk. Intuitively, a walk in a graph is the combinatorial analog of the continuous image of a closed real line segment, which may arbitrarily often cross or retrace itself, backward or forward. The formal definition of a walk is as follows: Definition 7 A walk W of length n from vertex u to vertex v of a graph G is an alternating sequence of vertices and directed edges, W = vo, e*1, nx,..., vn- 1 } efn,vn where v q = u, vn — v, and each edge is incident on the two vertices immediately preceding and following it, i.e., ai (BEGIN) = Vi-i, U i (END) — Vi. In addition, if vq = vn, the walk is closed; otherwise, it is open. The walk is a trail, if all its edges are distinct. The walk is a path, if all its vertices are distinct. A closed walk of length n > 3 with distinct vertices (except v$ = vn) is designated as a cycle. A walk W = v0, e^1, v \,..., ura_i, e£n, vn can be denoted as (v0, Vi)... (vn-i,vn) . D efinition 8 A graph G is connected, if for every pair of vertices u.v £ V (G) , then there exists a path in G joining u to v. A component of G is a maximal connected subgraph of G. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 12 Two graphs that can be described by the same picture are isomorphic. To be more precise, D efinition 9 Two graphs G\ and G2 are isomorphic (Gi = G f) , if there exists a bijective map 6 : Vqx — * V g2 preserving adjacency, that is, {u, n} G Igx if and only if{B(u),6(v)} G Iq2. Isomorphic graphs have the same degree. Therefore, the degree is an invariant. Given two graphs G\ and G2 with VgxC)Vg2 — 0, several operations that generate new graphs from Gi and G2 are defined as follows: D efinition 10 1. The union G = G\ U G2 is such that Vq = Vgx U Vq2 and Eg — Eqx U Eq2 ■ 2. The join (suspension) G = G\ + G2 is such that Vq = Vgx U Vq2 and Eg = Eqx U Eq2 U (Vg1 x Vq2) , where the endpoints of an edge e = (u, v) G (Vgj x Vq2) are the vertices u G Vqx and v G Vq2. 3. The Cartesian product G = Gi x G2 is such that Vq = V g x x Vq2 and E q = { E g x x V g 2) U (V g 1 x Eq2) , where the endpoints of an edge (e,v) G (E di x Vg2) are (u\,v) and {u2, v) with V g x (e) = {ax, ^2} , and the endpoints of an edge (u, / ) G (Vgx x Eq2) are (u, v i) and (u, v2) with V a2 (/) = {vi, v2} . The composition (lexicographic product) G = Gi [Gf\ is such that Vq = Vdi x V g2 and E g = ( E a x x Vg2 x Vg2) U (V g 1 x E g 2) 1 where the endpoints of an R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 13 edge (e,vi,v2) G (EG l x VGi x Vg2) are (ui,vi) and (u2,v2) with VG l (e) = {ui, ^2} , and the endpoints of an edge (u, /) e (Va, x £ ? G f2) o,re (u, Vi) and (u, v2) with Va2 (/) = {vi,v2} . 5. The edge-complement G of a graph G is such that VG = Vq and e g = {(^1^ 2) G Vg x VG | (ui, v2) < £ Vq (e) for e G £ G} • Several infinite families of graphs can be defined as follows: D efinition 11 1. A tree is a connected graph with no cycles. 2. A complete graph K n is a graph on n vertices such that every two vertices are adjacent (i.e., all (”) possible edges are present). 3. A totally disconnected (or empty ) graph K n is a graph consisting of n vertices such that E xn — 0. 4. A complete bipartite graph K m^ n is a graph of order m + n such that Kmtn — K m + K n. 5. A complete n partite graph Kp 1) P 2j...1 pn is a graph of Pi vertices such that E pi,p2,...,pn = KP i + K p2 + • • • + EP n . 6. An n-cube Qn is a graph that is defined recursively as Qi = k 2, Qn — E 2 x n ^ 2 . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 14 2.2 Topological surface In this section, the topological concept of surface and the classification of compact surfaces are discussed. The material in this section follows [Mas91], closely. The topological concept of a surface is a mathematical abstraction of the idea of a surface made of pieces of paper glued together. A surface is a topological space with the same local properties as a plane of Euclidean geometry. D efinition 12 A surface is a connected 2-dimensional manifold i.e., a Hausdorff space in which every point has an open neighborhood homeomorphic to the open 2-dimensional disc U2 = {x &W2 : \x\ < 1}. A surface is orientable, if every closed path is orientation preserving (i.e., the orientation is preserved by traveling once around the closed path). A surface is nonorientable, if it is not orientable. An example of a compact orientable surface is the 2-sphere iS '2 ={x € K3: |x| = 1} ; another important compact orientable surface is the torus, which can be described as a surface that is homeomorphic to the surface of a doughnut. An example of a compact nonorientable surface is the real projective plane, which can be described as a surface that is homeomorphic to the quotient space of the 2-sphere S2 obtained by identifying every pair of diametrically opposite points. A connected sum of two disjoint surfaces Sa and Sb, denoted by Sa # Sb, is formed by cutting a small circular hole in each surface and by gluing the two surfaces together along the boundaries of the holes. To be precise, subsets Da C Sa and R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 15 Dh C Sb are chosen such that Da and Db are homeomorphic to a closed disc. The complements of the interior of Da and Db are denoted by S'a and S ’ b , respectively. Given that h : dDa — > dDb is a homeomorphism, then the connected sum Sa # Sb is the quotient space of S'a U S'b obtained by identifying the point x and h (x ) for all points x in the boundary of Da. The classification theorem for compact surfaces can be stated as follows: Theorem 1 (classification of compact surface) Every compact surface is home omorphic to a sphere, to a connected sum of tori, or to a connected sum of projective planes. Proof. See [Mas91, Chap. 1, Thm. 5.1 ]. ■ From this theorem, every compact surface can be represented by a polygon subject to gluing whose symbolic representation is of the following form: 1. The sphere: aa~l 2. The connected sum of g tori: a ib ia ^ b ^ a ^ c d f1^ 1... agbga~lbfl 3. The connected sum of k projective planes: a ia ^ o ^ ... a^afe. Polygon representations of sphere, torus, and projective plane are shown in Figure 2.2. The connected sum of a torus and a projective plane is homeomorphic to the connected sum of three projective planes. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 1 6 a sphere torus projective plane Figure 2.2: Polygon representations of sphere, torus and projective plane. D efinition 13 A triangulation of a compact surface S consists of a finite family of closed subsets {Tj, T2,. . . , Tn} that cover S and a family of homeomorphisms Pi : Ti — + Ti,i — 1,2,... ,n where each Ti is a triangle in the plane M2.The image of the vertices under p{ is designated as vertex set and the image of the edges under Pi is designated as edge set. In addition, it is required that any two triangles Ti and Tj, either be disjoint, or have a single vertex in common, or have one entire edge in common. From the strong form of the Jordan curve theorem, there exists such a trian gulation for any compact surface S. Therefore, the Euler characteristic x (S) for a compact surface S with triangulation {Ti,T2, ..., Tn} can be defined as X(S) = v ~ e + f (2.1) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 17 where v = total number of vertices of S, e — total number of edges of S, f = total number of triangles (= n ) . In fact, the Euler characteristic depends only on the surface S, not on the tri angulation chosen. In addition, the subdivision of S can be allowed into arbitrary polygons such that the interior of each polygon is homeomorphic to an open disc and the closure of each edge is homeomorphic to a closed interval or a circle. Fi nally, the Euler characteristic is a topological invariant and does not depend on the subdivision of S into polygons. Therefore, the Euler characteristic of a surface S can be redefined as follows: D efinition 14 For a compact surface S, the Euler characteristic x (S) is defined as x (S) = v - e + f (2 .2 ) where v — total number of vertices, e = total number of edges, f — total number of regions or faces, and v, e, f are obtained from arbitrary subdivision of a surface S into polygons. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 18 surface Euler characteristic Sphere 2 connected sum of n tori 2 — 2 n connected sum of n projective plane 2 — n Table 2.1: Euler characteristic of various compact surfaces The Euler characteristic of the connected sum Sa Sb can be computed by the following formula: x(Sa # S b) = x(Sa) + x ( S b) - 2 (2.3) The connected sum of n tori can be viewed as a sphere with n handles and can be denoted by Sn. Similarly, the connected sum of n projective planes can be viewed as a sphere with n crosscaps and can be denoted by Nn. The Euler characteristics of particular surfaces are shown in Table 2.1. Theorem 2 Given that Sa and Sb are compact surfaces, then Sa and Sb are home omorphic if and only if their Euler characteristics are equal and both are orientable or both are nonorientable. Proof. See [Mas91, Chap. 1, Thm. 8.2]. ■ D efinition 15 The genus g (S) of a surface S is the number of handles (for an orientable surface) or crosscaps (for a nonorientable surface) that one must add to the sphere to obtain a surface that is homeomorphic to the surface S. A surface that R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 19 handle sphere Figure 2.3: Embedding graph into a sphere by pulling handle. is a connected sum of n tori or n projective planes is said to be of genus n. A sphere is of genus 0. Hence the genus { I (2 — x (S)) in the orientable surface case (2.4) 2 — x (S) in the nonorientable surface case. 2.3 Embedding of a graph into a surface The embedding of a graph G into a surface S is how to draw a graph G on a surface S without edge crossings. The embedding of a graph into a surface can be obtained by the following procedure: Draw the graph on the sphere S2 possibly with some links crossing; for every crossing, attach a handle to the sphere S 2 such that one of the links passes through the handle rather than the sphere (this procedure removes the crossings once at a time); after removing all crossings by attaching handles, the graph is embedded in a surface S with g handles. Now, the formal definition of embedding a graph G in a surface S is as follows: D efinition 16 A graph G is embedded in a surface S, if it is drawn in S so that R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 0 edges intersect only at their common vertices. The components of S — G are regions (or faces) of the embedding. A region is a 2-cell, if it is homeomorphic to the open unit disk. In addition, if every region of an embedding is a 2-cell, the embedding is a 2-cell embedding. In addition, a region is a 2-cell, if any simple closed curve In this region can be continuously deformed or contracted into a single point. Definition 17 The genus g (G) of a graph G is the minimum genus among the genera of all surfaces S in which G can be embedded. An embedding of a graph G in the compact surface Sk is a minimal embedding, if g (G) = k. Theorem 3 Given that G is a connected graph embedded in a surface of genus g which is equal to the genus of the graph, then every region of G is a 2-cell and the embedding is a 2-cell embedding. Proof. See [WhiOl, Thm. 6-11]. ■ Definition 18 The maximum genus gM {G) of a graph G is the maximum genus among the genera of all orientable surfaces in which G can be 2-cell embedded. Definition 19 A graph is planar, if it can be embedded in the plane (or equiva lently, in the surface So by the stereographic projection). If G is planar, then g (G) = 0. If g (G) = k, k > 0, then G has an embedding in Sk, but not in Sh, for h < k. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. iL K. 3,3 Figure 2.4: The nonplanar graph K$ and 1^3,3 ■ Theorem 4 (K uratow ski) A graph G is planar if and only if it contains no sub graph that is homeomorphic to either K 5 or K$^. Proof. See [GT01, Sec. 1.4.5 ]. ■ The complete graph K 5 and the complete bipartite graph are called Kura- towski’s graphs. D efinition 20 Given that G is a connected graph with a 2-cell embedding on an orientable surface S, then the Euler characteristic of a 2-cell embedding G — > S, x (G — > S') is defined by x(G->S) = |r| —|£j + m (2.5) where V is the set of vertices, E is the set of edges, and F is the set of regions. Therefore, the Euler characteristic of a 2-cell embedding is equal to the Euler characteristic of surface S, i.e., X(G-*S) = X (S) 2 - 2g, if S = Sg 2 - k , if S = Nk (2.6) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 2 T heorem 5 For each orientable surface Sg (g = 0,1,2,...), there exists a con nected graph G and a 2-cell embedding G Sg whose Euler characteristic satisfies the equation x (G) — 2 — 2 g. For each nonorientable surface (k = 0, 1, 2,...), there is a graph G and a 2-cell embedding of G into the surface Nk such that X (G) — 2 — k. Proof. See [GT01, Thm. 3.3.1 and Thm. 3.3.2]. ■ 2.4 Algorithm for minimum embedding An important problem in graph theory is how to determine the genus of a graph. This problem can be translated into a combinatorial problem of determining a rotation system with the maximum number of regions. Intuitively, each rotation system can be considered as an algebraic description of a 2-cell embedding. The vertex set of a connected graph G can be denoted by Vq = {1, 2, ,n} . Given that V (i) = {k E Vq | {i, k} E Eg} for each i E Vq, then define Pi : V (i) — > V (i) to be a cyclic permutation on V (i), of length n, = \V (i)|; pi is designated as the rotation at i and the set {pi,P2 , ■ • - ,Pn} of rotations is designated as a rotation system, or rotation scheme. T heorem 6 Every rotation system {pi,P2, ■ ■ • ,Pn} for a graph G determines a 2-cell embedding of G into an oriented surface S, such that the orientation on S induces a cyclic ordering of the edges {i, k} at i in which the immediate successor to R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 23 {*, k} is (k)} , i = 1 ,..., n. In fact, given {pi,P2, ■ ■ ■ ,Pn} , there is an algorithm which produces the embedding. Conversely, given a 2-cell embedding in a surface S with a given orientation, there is a corresponding {p i,p 2 , • •. ,pn} determining that embedding. Proof. See [WhiOl, Thm. 6-50 ]. ■ Given E — {(a, b) | {a, b} £ Eg} , and P a permutation on the set E of di rected edges (where each edge of G is associated with two possible directions) such that P (a, b) = (b,pb (a) ) , then an orbit under P is a closed walk W = (i0, if) ■ ■ ■ (im-i,im ) with the following properties: 1. For every j ± k, (ij-i,ij) ^ {h-iPk) • 2 . (fkt ik+i) = P if) j k = 1, 2 , .. ., m — 1 3. (Iqj if) P im). Notice that (ij,ik) 7^ (ik, ij) since they consists of different directions. Then each orbit under P determines a 2-cell region of the corresponding embedding. Hence the number of orbits is the number of faces of the 2-cell embedding. Finally, regions can be pasted together with (a, b) matched with (b, a) to obtain an orientable surface. Since an embedding of G into Sg(a) is a 2-cell embedding, there is a rotation system corresponding to this 2-cell embedding. It now follows that the genus of n any connected graph can be computed by selecting among the (n* — 1)! possible i— 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 24 rotation systems the one that gives the maximum number of orbits, and hence determines the genus of a graph. The maximum genus of a graph can be computed from a rotation system with minimum number of orbits. To illustrate the concept of rotation system, three rotation systems of the com plete graph K5 are considered. Let VK h = {1,2, 3,4,5} with V (i) = Vk5 — {«} . Define a rotation system by Pi ■ (2,3,4,5) P2 : (3,4,5,1) Ps : (4,5,1,2) P i ■ (5,1, 2,3) Ps ■ (1,2,3,4) The orbits under this rotation system are 1. (1,2) (2,3) (3,4) (4,5) (5,1) 2- (1,3) (3,2) (2,4) (4,3) (3,5) (5,4) (4,1) (1,5) (5,2) (2,1) 3. (1,4) (4,2) (2,5) (5,3) (3,1) Prom this rotation system, we have x (Ks) = 5 — 10 + 3 = — 2. This implies that 2 — 2g = — 2, g = 2. This rotation system corresponds to an embedding of the graph K 5 into an orientable surface with genus 2. Now consider another rotation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 25 system defined by Pi ■ (5,4,3, 2) P2 : (1,4,3,5) Ps : (2,1,4,5) Pi ' ■ (5,3,2,1) Ps : (3,4,1,2) The orbits under this rotation system are 1. (1,2) (2,4) (4,1) (1,3) (3,4) (4,2) (2,3) (3,1) 2. (1,4) (4,5) (5,1) 3- (1,5) (5,2) (2 , 1) 4. (2,5) (5,3) (3,2) 5. (3,5) (5,4) (4,3) Hence this rotation system yields x (K*,) = 5 — 10 + 5 = 0. This implies that 2 — 2g = 0, g — 1. This rotation system corresponds to an embedding of the graph K§ into an orientable surface with the minimum genus of the graph K5. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 26 Consider the other rotation defined by Pi : (2,5,4,3) P2 ’ ■ (1,5,4,3) P3 : (1,5,4,2) Pa ■ (1,5,3,2) P 5 ' • ( 1 , 3 , 4 , 2 ) The orbits under this rotation system are 1. (1,2) (2,5) (5,1) (1,4) (4,5) (5, 2) (2,4) (4,1) (1,3) (3,5) (5,4) (4,3) (3,2) (2,1) (1,5) (5,3) (3,4) (4, 2) (2,3) (3,1). Hence this rotation system yields x {K < o ) — 5 — 10 + 1 = — 4. This implies that 2 — 2g = — 4, g = 3. This is the maximum genus of the graph K 5. Hence this rotation system corresponds to an embedding of the graph Jf5 into an orientable surface with maximum genus. In fact, among all 7776 possible permutations (((4 — l ) !) 5 = 7776), there are 462 rotation systems of genus one, 4974 rotation systems of genus two, and 2340 rotation systems of genus three. Table 2.2 shows the formula for some well known graphs [WhiOl, Chap. 6 , Sec. 6-4 and 6-5]. Note that [x\ denotes the greatest integer less than or equal to x\ \x\ denotes the least integer greater than or equal to x. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 7 Graph genus g maximum genus gu Q n 1 + 2n~3 (n - 4), n > 2 (n — 2 ) 2""2, n > 2 K n (n — 3 )(n — 4) 12 , n > 3 (n—l)(n~2) 4 K m ,n (t o — 2 )(n — 2) 4 , m ,n > 2 (m — l)(ro— 1) 2 Table 2.2: Genus and maximum genus of some families graphs The problem of computing the genus of a graph is known to be NP-complete [Tho89]. Therefore, there is no polynomial bounded algorithm for deciding the genus of graph. 2.5 Cayley graph The material in this section follows [GT01], [WhiOl], and [dlH O O ] closely. The concept of Cayley graph is to view a given finitely-generated group as a graph. D efinition 21 Given that S = {gi,g2, ■ ■■} is a nonempty subset of a group F, then a word W in S is a finite product / 1 / 2 ■ ■ ■ f n, where each f i £ { i?i, £ 2 , • • •, \ <?2 1> • • •} • If every element of F can be expressed as a word in S, then S is a generator set of the group F. A relation is an equality between two words in S. If F is generated by gi: 03) • • • ond every relation in F can be deduced from the relations P = P',Q = R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 28 Qf,R= R ',..., then a presentation ofT is {gi, 92,g3,---\P = P', Q = Q ',R = R',...) ■ A presentation is finitely generated if the number of generators is finite. A finite presentation is both finitely generated and finitely related. Definition 22 Given that T is a group and S is a generator set for T, then the Cayley color graph Cs (F) is a digraph such that Vcs(r) — F and -E 'C s(r) = { (< ? > gh) | g G F, h e S} . Definition 23 An automorphism of a Cayley color graph C$ (F) is a permutation 9 of Vcg(r) such that for every gx and g^ in F and h in S, gxh = c ? 2 if and only if 9 {gi) h = 9 (g2). Given that Aut (Cs (F)) is the collection of all automorphisms of Cs (r), then Aut (Cs (F)) forms a group with the group multiplication defined by the composi tion of automorphisms. In addition, the following theorem shows that Aut (Cs (F)) is isomorphic to F. T heorem 7 Given that Cs (F) is any Cayley color graph for the finite group F, then Aut (Cs (r)) = F (independent of the presentation selected for F). Proof. See [WhiOl, Thm. 4-8]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 29 D efinition 24 Given that S is a generating set for the group F subject to the following conditions: 1. e i S 2. If h £ S, h~l S (unless h2 = e). 3. If h £ S, h2 — e, then each pair (g, gh) and (gh, g) of directed edges are coalesced into a single undirected edge {g,gh} . Then the Cayley graph Gs (F) is the graph underlying the Cayley color graph CS (T). Definition 25 Given that F is a group, then the genus of the group F, g (F), is j( r ) = nM {j(C s (r))} (2.7) where the minimum is taken over all generating sets S for F. The genus of an infinite group is either zero or infinite. This result follows from Levinson’s theorem: Theorem 8 (Levinson) If T is an infinite group, with G the Cayley graph of a presentation for F, then either g (G) = 0, or g (G) = oo. Proof. See [WhiOl, Thm. 7-9 ]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 30 D efinition 26 Given a Cayley graph Gs (F), then the word length Is (g) of an element g G F is the smallest integer n such that g = (S1S2 ... sn) where S { € SU S^1. In addition, the word metric ds is defined on G by ds {gi, 92) = Is {9 ^ 92) , (2.8) for every gi,g2 G G. Therefore, the finitely generated group F can be considered as a discrete metric space, and the action o f go G F by left multiplication is an isometry. Each edge of Gs (F) can be considered as a metric space that is isomorphic to the segment [ 0 , 1] of the real line, in such a way that the left action of G produces isometries between the edges. The distance between two points in a Cayley graph is the infimum of the path-lengths joining these points. Therefore, the Cayley graph can be considered as an arc-connected metric space. 2.6 Weighted directed graph In many real problems, the many edges of a graph are metrically unequal. This phenomenon is reflected in the following definition. D efinition 27 Given that G is a graph, then a weighted graph is a pair (G, w) where a weight function w : Eg — > • R maps edges to real-valued weights. The weight of a path p = (vo, tq)... (fn_i, vn) is the sum of the weights of its constituent edges. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 31 n That is, w (p) = w (wj_i, W j). In addition, the shortest path weight from u to v i=1 where u ,v E Vg is defined as follows. min {w (p) | p is a path from u to v} if there is a path from u to v, d (u , v) oc otherwise. (2.9) A trivial example of a weighted directed graph is a directed graph with constant weight function. A Cayley graph is a weighted graph where each edge has unit weight and the shortest path weight from gi to g2, where g i,g 2 6 F, is equal to word metric ds {gi,g2) • Given that (G , w) is a weighted directed graph, then the single-source shortest paths problem is to find a shortest path from a given source vertex s E Vq to all other vertices v E Vq . One example of the single-source shortest path problem is the routing in the internet. In this problem, the internet can be considered as a weighted directed graph where the weight function can be obtained from several metrics in the communication links such as distance, cost, capacity, delay, number of hops. Routing is the process of finding a shortest path from source to destination. Each router will make, locally, routing decisions, but based on global information. This routing decision is done using Routing Protocols located in the network (or the IP) layer. The solution of single-source shortest paths can be classified into two categories: Dijkstra algorithm and Bellman-Ford algorithm. Both algorithms have been used R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 32 In routing algorithms in the internet (see [Sta98] and [StaOO]). The analysis of these two algorithms can be founded in [CLR90]. The Dijkstra’s algorithm has been implemented in link-state based routing algorithm and the Bellman-Ford algorithm has been Implemented in distance-vector based algorithm. In the link-state based routing algorithm, each router will broadcast other routers in the network the state of its immediate links. Based on this informa tion, each router in the network will make its routing decision. An example of link-state based algorithm is open shortest path first (OSPF). In the distance-vector based algorithm, each router will inform all Its neighbor ing routers of the distances between this router and all other routers in the network. Based on this information, each router in the network will make its routing deci sion. This algorithm obviously does not scale well with the size of the network. An example of distance-vector based algorithm is routing information protocol (RIP). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 33 Chapter 3 Riemannian Geometry In the previous chapter, the concept of embedding a graph into a topological surface was discussed. In this chapter, the concept of Riemannian geometry is discussed. This chapter involves the following: First, the basic geometric concepts, such as differentiable manifolds, tangent spaces, vector fields; second, Riemannian metrics; third, Riemannian connections; next, geodesics, curvature and Jacobi field; finally, hyperbolic spaces with constant curvature. The material in this chapter follows [Jos02], [Car92], and [Boo02] closely. 3.1 Differential manifolds The concept of differentiable manifold is the abstract idea of extending differential calculus to a more general space than Rn. The formal definition of differentiable manifold is given as follows: D efinition 28 An n-dimensional differentiable manifold M of class Ck is a second countable Hausdorff topological space M with a family of injective mappings xa : R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 34 Ua — > R" from open sets Ua C M into Rn such that 1. UaUa = M. 2. For any pair of a, j3 with Ua fl Up 0, xa (Ua) and xp (Up) are open sets in Rn and the mapping xp o x ff : xa (Ua fl Up) — ► xp (Ua fl Up) is differentiable of class Ck. 3. The family {(Ua,x a)} is maximal relative to conditions 1 and 2. The mapping xa is called a coordinate map and a pair (Ua, xa) is called a coordinate system. A family {(Ua,xa)} that satisfies 1 through 3 is called a differentiable structure of class Ck on M. D efinition 29 Given that M is a differentiable manifold, then M is orientable if M admits a differentiable structure {(Ua,xa)} such that for every pair of a, ft with Ua P i U p 0, the differential of the change of coordinates xp o x~l has positive determinant. Otherwise, M is nonorientable. D efinition 30 A mapping h : M N between differentiable manifolds M and N with coordinate systems {(Ua,x a) : a e A} and {(Up,xp) : { 3 £ B} is differentiable if for each pair of a € A, (3 £ B the map xp o h o x ff is differentiable wherever it is defined. A map h : M N is a diffeomorphism if it is bijective and both h and h~l are differentiable. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 35 The tangent space to a manifold is one of the most important ideas in dif ferential geometry. This is based on the intuitive idea of the tangent plane to a surface. However, the latter involves embedding the manifold into a higher di mensional vector space. Developing an idea of a tangent space that is intrinsic to the manifold itself relates to the local differentiable properties of functions on the manifolds. There are two different approaches [Ish99, Sec. 2.3], namely, geometric and algebraic, to define the tangent space. These two approaches are different in an infinite-dimensional manifold. However, in finite-dimension, both definitions are equivalent [War83, Sec. 1.21]. In the geometric approach, the tangent vector at a point can be considered as an equivalent class of curves tangent to each other at that point on the manifold and can be defined as follows: Definition 31 Given that M is a differentiable manifold, then a differentiable curve in M is a differentiable function a : (— e, e) — > M. Two curves cq and a 2 are tangent at a point p in M if cq (0) = cq (0) = p and dxl . .... dxl . . . . . . ( « i (*)) |t= o= («2 (*)) |t=o (3.1) in some local coordinate system (x1, ..., xn) that covers the point p. A tangent vector at p € M is an equivalence class of curves in M where two curves are in the same equivalence class if they are tangent to each other at the point p. The tangent space TP M to M at point p is the set of all tangent vectors at the point p which R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 36 carries a structure of real vector space. In the algebraic approach, the tangent vector can be considered as a directional derivative and can be defined as follows: D efinition 32 Given that V is the set of functions on M that are differentiable at p, then a derivation at a point p £ M is a map v : V — > E such that 1. v ( / + A g) = v(f) + Xv (g), for all f,g £ V , A e l ; v(f -g) = f {p) v (g) + g (p) v (/), for all f,g £V. The set of all derivations at p £ M is denoted by DP M and can be given the structure of a real vector space by defining (v + w) (/) : =v(f) + w (/) (3.2) (Av)(f) : = Xv (/) (3.3) for all v, w £ DP M and A g I . The following theorem shows that TPM and DP M are isomorphic as vector spaces: T heorem 9 The linear map i : TPM — > DPM defined by df (a (t)) i (v)(/) := dt (3.4) t= 0 where v £ TpM, a is a curve in the equivalence class of v, and f £ V is an isomorphism. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 7 P roof. See [Mi99, Thm. 2.2]. ■ D efinition 33 The tangent bundle T M is the set {(p, v) : p £ M, v £ TPM} with a differentiable structure of dimension 2n. A vector field X on a differentiable manifold M is a correspondence that associates to each point p £ M, a vector X (p) £ TP M. A vector field is differentiable if the mapping X : M T M is differ entiable. A vector field V along a curve 7 : [a,b] M is a differentiable mapping that associates to every t £ [a, 6 ] C R a tangent vector V (t) £ T ^ M . The velocity field of 7 is the vector field d j (^ ) := D efinition 34 Given that X ,Y are vector fields of class C°°, then the Lie bracket [X, Y] is the vector field X Y — Y X . The vector fields X andY commute if [X, Y] = 0 . 3.2 Riemannian manifolds A Riemannian metric is a metric structure on a differentiable manifold and can be defined as follows: D efinition 35 A Riemannian metric on a differentiable manifold M is a corre spondence which associates to each point p of M an inner product ( , )p on the tangent space TP M which depends smoothly on the base point p. A Riemannian manifold is a differentiable manifold, equipped with a Riemannian metric. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 38 In general, each differentiable manifold can be equipped with a Riemannian metric and in each local coordinate O S — j O C ■ ) . . p > ^ X | i ) a Riemannian metric can be represented by a positive definite, symmetric matrix where the coefficients depend smoothly on x. Hence the inner product of two tangent vectors v,w G TP M where v = (n1, ..., vn) and w — (w1, ..., wn) is given by where the Einstein summation convention is used, that is, an index occurring twice, once covariantly once contravariantly, in a product is to be summed from 1 to n (the dimension of the manifold M). D efinition 36 Given that 7 : [a, 6] — > M is a differentiable curve in a Riemannian manifold M , then the length of 7 is given by n n (v,w) = 9ii (?))V % V )3 := dij (x ip)) tfw3 i= 1 j=l (3.5) b a (3.6) a and the energy of'y is given by (3.7) a R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 39 The length and energy of 7 can be expressed in local coordinates — (''T — 1 \ b a > * ^ j ^ e > 0 6 by the following formula: b 1 (7 ) = j J g v (x ( 7 (())) i ‘ (i) iJ (t)dt, (3.8) a b E(l) = i j ( t ) i ? (t)dt (3.9) a The length of a piecewise differentiable curve is the sum of the lengths of the smooth pieces. Similarly, the energy of a piecewise smooth curve is the sum of the energies of the smooth pieces. On a Riemannian manifold M, the distance between two points p, q can be defined as follows: d(p, q) ==inf{L(7 ) : 7 : [a, b] — »M piecewise differentiable curve with 7 (a) =p, 7 (h)=q}. It is clear that this distance function satisfies the following axioms: 1 . d (p, q) > 0 for all p, q, and d (p, q) > 0 for all p 7^ q, 2 . d(p,q) = d(q,p), 3. d (p, q) < d (p, r) + d (r, q) (triangle inequality) for all points p,q,r e M. Hence a Riemannian manifold M can be considered as a metric space with a distance d. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 0 3.3 R iem annian Connections D efinition 37 Given that X (M ) is the set of all vector fields of class C°°, then an affine connection V on a differentiable manifold M is a mapping V : X ( M ) x X (M) -+ X (M) which is denoted by (X, Y ) — > V x Y and which satisfies the following properties: 1- ^ f x + gy Z = f X x Z + gXyZ. 2. Vx (Y + Z) = V XY + V XZ. 3. Vx (f Y ) = f V x Y + X (/) y, in which X,Y,Z E X (M) and f,geV(M). Definition 38 An affine connection V on a differentiable manifold M is symmetric if for every X ,Y E X (M), VxY - X yX = [X,Y}. (3.10) Definition 39 Given that M is a differentiable manifold with an affine connection V, V is a vector field along the differentiable curve 7 : I — > M, then the covariant derivative of V along 7 is a vector field along 7 that satisfies the following properties: 1 D (V + W \ = dv + dw d t \ v ^ v v ) dt ^ dt ■ 2. (fV) = + f^jf, where f is a differentiable function on I. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 41 3. If V is induced by a vector field Y £ X (M), i.e., V (t) = Y (7 (£)), then ¥ = v ^ y . ai dt Theorem 10 (Levi-Civita) Given a Riemannian manifold M, then there exists a unique affine connection V on M satisfying the conditions: 1. V is symmetric. 2. V is compatible with Riemannian metric, that is, for every X,Y,Z £ X (M), X((Y,Z)) = (Vx Y,Z) + (Y,Vx Z). Proof. See [Cax92, Chap. 2 Thm. 3.6]. ■ Definition 40 The affine connection in the previous theorem is called the Levi- Civita (or Riemannian) connection and can be determined by the following formula (VXY,Z) = ±(X(Y,Z)-Z(X,Y) + Y(Z,X)) - (x, [y, zj) + (z, [ x ,y ]) + <y, [z,x\) . (3 .1 1) In a coordinate chart (U, x ) , the Levi-Civita connection is given by 8 8 V jlt— = r t ~ (3.12) ax' dx? 3 8xk where the F^.’ s are called the Christoffel symbols. The expression for the Christoffel symbols is given by the following formula: Fjfc ~ lfi9 (dlk,j + 9jl,k ~ 9jk,l) 1 (3.13) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 42 where (gij) . . , is the inverse matrix of (gij)i^_1 n .and d 9jk,i (a;) = g - 9jk (x) . (3.14) Given that V = vl-£x, then ^ (3-15) dt V dt 1 3 dt j dxk' 3.4 Geodesics Definition 41 Given that M is a differential manifold with an affine connection V, then a vector field V along a curve 7 : I — > M is parallel to itself if = 0, for all t £ I. A parameterized curve 7 : I — + M is a geodesic if the vector field ^ is parallel to itself along the curve 7 i.e., D { d'y In a coordinate system (U,x), 7 (t) = {xl (t),..., xn (t)) is a geodesic if and only if D fd'y'j d dt \ dt = (#* (t) + T)k (x (t)) xj (t) xk (t)) ^ 0 = x% (t) + F*f c {x (t)) x3 (t) xk (t), i = 1,2,... ,n. (3.17) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 43 Given that 7 : [a, b ] — > M is a geodesic and that the connection is compatible with the metric, then d_/dry dsy\ = 2 / — — — \ = 0 d t \ d t , dtf \dt dt' dt / ’ L (7 ) = c (6 — a ) . (3.18) D efinition 42 vl curve 7 i s parameterized proportionally to arc-length if ||^|| = constant almost everywhere. In addition, if ||^|| = 1, the curve 7 is parameterized by arc-length. Hence a geodesic is always parameterized proportionally to arc-length. In fact, there exists a diffeomorphism ip : [0, L (7 )] — > [a, b ] such that d j o ip 1 dt for almost every t. Hence a geodesic 7 can be reparameterized such that 7 : [0, L (7 )] — ► M is parameterized by arc-length. Lem m a 11 J/ 7 : [a, b ] — > M is a differentiable curve, and ip : [a, 0\ -> [u, b \ is a change of parameter, then L ( 7 0 ip) — L (7 ) ■ (3.19) Proof. See [Jos02, Chap. 1, Lemma 1.4.3]. ■ Prom lemma 11, the length of a differentiable curve is invariant under a change of parameter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 4 Lem m a 12 For each differentiable curve 7 : [a, b ] — > M L (q) 2 < 2 (b — a )E (7 ) (3.20) and equality holds if and only if ||^|| = constant. Proof. See [Jos02, Chap. 1, Lemma 1.4.3 ]. ■ Therefore, a differentiable curve 7 that is parameterized by arc-length has the smallest energy among all parameterized curves defined over [0, L (7 )] and has 2E (7 ) = \L (7 ). Therefore the problem of finding the shortest length curve be tween two points is equivalent to the problem of finding the curve that minimizes the energy E. The Euler-Lagrange equations for E are given by the following equation (see [JLJ98, Part 1, Chap. 2]): (3.21) where x'(t) = xl ( 7 (t)). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 45 Then ^ (9ik (x (t)) xk (t) + gjt (x (£)) x3 (£)) = ^ f c i i (x (£)) id (t) xk (t) 9ikxk + gjiX3 + gik,ixlxk + gji,ixlx3 = gjktix3xk gimxm + ^ (gikj + gji,k - & •*,:) = o gllgimxm + ^ (gikj + gji,k - % /= ,i) = 0. (3.22) Hence x 1 (t) + r*-f c (x (£)) id (£) xf c (f) = 0, £ = 1,2,..., n. (3.23) That is, the geodesic is the solution of the Euler-Lagrange equations for energy E. T heorem 13 A geodesic ^ on a Riemannian manifold M satisfies the Euler-Lagrange equations for the energy E. Proof. See [Jos02, Chap. 1, Lemma 1.4.4]. ■ The geodesic equation is in general a nonlinear ordinary differential equation. From the existence and uniqueness of differential equations, the following theorem can be proved. Theorem 14 Given p E M, and v E TP M where M is a Riemannian manifold, then there exists e > 0 and a unique geodesic 7 : (— e, e) — > M that depends smoothly on p and v such that 7 (0) = p and 7 (0 ) = v. Proof. See [Jos02, Chap. 1, Thm. 1.4.2]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 6 Denoting the unique geodesic of Theorem 14 by 7 with t E (—e,e) , 7 (0) — p, and 7 (0) — v by 7 (t,p,v), then the geodesic 7 (f, q, Xv), A G 1 is defined on interval (— f,f) and Hence the geodesic is homogeneous. Since 7 (t,p,v) depends smoothly on v and {v E TPM : |M| = 1} is compact, then there exists £0 such that for any t E [0, £0] and v E TPM with ||u|| = 1, 7 (t,p,v ) is well defined. Therefore, for any t E [0,1] and v E TPM with ||w|| < e0, 7 (t,p,v) is well defined. Then the exponential map can be defined as follows: D efinition 43 Given that M is a Riemannian manifold, p E M, and Vp — {vE TPM : 7 (t,p, v) is well defined on [0,1]} , then the map expp : Vp — ► M given by is called the exponential map of M at p. Theorem 15 Given p E M, then there exists a neighborhood W of the origin in TP M such that the exponential map expp : W C TP M — » M is a diffeomorphism onto a neighborhood of p E M. Proof. See [Jos02, Chap. 1, Thm. 1.4.3]. ■ 7 (t,p, Xv) = 7 (Xt,p,v). (3.24) (3.25) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 47 D efinition 44 Given that p £ M, and U = expp W where W is defined as in Theorem 15, then U is called a normal neighborhood of p and the local coordinates defined by a pair (U, exp"1 ) is called (Riemannian) normal coordinates system with center p. In addition, if Be (0) is such that Be (0) C W, then expp Be (0) := Be (p) is called the normal ball (or geodesic ball) with center p and radius e. T heorem 16 In normal coordinates system, the Riemannian metric and Christof fel symbols are 9ij (0) = dij, (3.26) rjfc (0) = 0 for alii, j,k . (3.27) Proof. See [Jos02, Chap. 1, Thm 1.4.4]. ■ The following theorem shows that geodesics locally minimize the arc length. T heorem 17 Given p £ M, U a normal neighborhood of p, B C U a normal ball of center p, if 7 : [0,1] — > B is a geodesic that joins 7 (0) = p and 7 (1), then L (7 ) < L (c), where c : [0,1] — » M is any piecewise differentiable curve joining 7 (0 ) to 7 (1). Proof. See [Car92, Chap. 3, Prop. 3.6]. ■ The following theorem formulates the converse of Theorem 17. Theorem 18 If j : [a,b] — » ■ M is a piecewise differentiable curve parameterized proportionally to arc-length which has length less than or equal to the length of other piecewise differentiable curve joining 7 (a) to 7 (b), then j is a geodesic. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 48 Proof. See [Car92, Chap. 3, Corollary 3.9]. ■ D efinition 45 A Riemannian manifold M is (geodesically) complete if for all p G M, the exponential map expp is defined for all v £ TP M, i.e., any geodesic 7 (t) with 7 (0 )=p is defined for all 1 G l. The Hopf-Rinow theorem can be stated as follows: T heorem 19 (Hopf-Rinow for R iem annian m anifold) Given that M is a Rie mannian manifold and p £ M, then the following statements are equivalent: 1. expp is defined on all of TP M. 2. M is geodesically complete. 3. The closed and bounded subsets of Ifil are compact. 4- M is complete as a metric space. 5. There exists a sequence of compact subsets K n C M, K n C K n+ 1 and UnKn = M, such that if qn ^ K n then d (p, qn) — > 0 0 . In addition, each of the statements above implies that for any two points p,q £ M, there exists a geodesic 7 joining p to q with Lfif) = d (p, q) . Proof. See [Jos02, Chap. 1, Thm. 1.4.8]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 49 3.5 Curvature D efinition 46 The curvature R of a Riemannian manifold M is a correspondence that associates to every pair X ,Y E X (M) a mapping R(X,Y) : X (M) — » X (M ) given by R (X, Y) Z = X y V x Z - V xV yZ + V [X,Y]Z, Z E X (M), (3.28) where V is the Riemannian connection of M. P ro p o sitio n 20 Given that X , Y, Z, W E X (M ), and f,g E V (M ), then the curvature R of a Riemannian manifold M has the following properties: 1. R ( fX + gZ, Y) = f R (X, Y) + gR (Z, Y ) , - 8. R (X, f Y + gZ) = f R (X, Y) + gR (X, Z ) , 3. R(X,Y) (Z + W) = R(X,Y) Z + R(X,Y)W, 4. R(X,Y) f Z = f R (X, Y) Z. Proof. See [Car92, Chap. 4, Prop. 2.2]. ■ Given that V is a vector space, then the area of a two-dimensional parallelo gram constructed on the pair of vectors x, y E V is denoted by \x A y\ and can be computed by (3.29) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 50 L em m a 21 Given thatp G M is a point, a C TP M is a two-dimensional subspace, and {x,y} ,{z,w} are two basis of a, then (R{x,y)y,x) (R(z,w)w,z) /0 on, I . >2 — i . .2 (O.oUj \x/\y\ \zAw\ Proof. See [Car92, Chap. 4, Prop. 3.1]. ■ The sectional curvature can be defined as follows: D efinition 47 Given a point p & M and a two-dimensional subspace a C TP M, then the sectional curvature of the manifold M at p relative to the subspaces a is (R (x,y)y,x) \x A y\* k ( ct) — ; (3.31) where x, y are two linearly independent vectors in a. Definition 48 Given the two functions 5, A : M — > R defined by 5 (p) = min k (a), (3.32) ( 7 A (< r) = max k (a), (3.33) <7 where the maximum and minimum are taken over all two-dimensional subspace of TP M, then the Riemannian manifold M is 1. of positive curvature if 5 (p) > 0 for all p, 2. of negative curvature if A (p) < 0 for all p, 3. of constant curvature if 6 — A is constant, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 51 4- flat if 8 — A = 0. D efinition 49 Given that p € M, that {z\, z%,..., zn} is an orthonormal basis of TP M, then the Ricci curvature in the direction Zi is Ricp (Zi) = — Y (R (Zi, zfl zu Zj), (3.34) n — I z— * ' 3 where j = 1,..., n and j i. In addition, the scalar curvature at p is n k(p) = - Y Ricp (Z i) n i— \ ~ n (n — l) ^ h ) (3.35) and it is independent of a choice of orthonormal basis of TP M. Definition 50 Given that 7 : [0,1 ] — ► M is a curve parameterized by arc-length in M, then the covariant derivative of 7 along the curve 7 at p is called the geodesic curvature of 7 at p. The local and global Gauss-Bonnet theorems can be state as follows: Theorem 22 (Local Gauss-Bonnet) Given that M is an oriented 2-dimensional Riemannian manifold with sectional curvature k and volume element dA, and N C M is a polygon which is diffeomorphic to a subset of R2 such that ON has vertices R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 52 at ti,.. ., tn with discontinuity 8%,... ,9n and Kg is its geodesic curvature with arc length ds, then [ ndA + f Kgds + V 6i = 27r. (3.37) J JdN N * — 1 Proof. See [Car76, Sec. 4-5]. ■ Theorem 23 (Global G auss-B onnet) Given that M is an oriented 2-dimensional Riemannian manifold with sectional curvature k and volume element dA, and R C M is a regular region of M such that dR has vertices at t \ ,..., tn with discontinuity 9i,... ,9n and k 9 is its geodesic curvature with volume element ds, then f ndA + [ Kgds + 'S'' 9i = 2irx (R) ■ (3.38) J J dR R % ~ x Proof. See [Car76, Sec. 4-5]. ■ Corollary 24 Given that M is an orientahle compact surface, then f ndA — 2 7rx (M). (3.39) Jm Proof. See [Car76, Sec. 4-5, Cor. 2]. ■ C orollary 25 Given a geodesic triangle A {A, B, C) with interior angle a, j3,7 in a Riemannian manifold with constant sectional curvature k , then 1. if k < 0, then area of A (A, B, C) = ~~(2±S±Jl • 2 . if k — 0 , then a + /? + 7 = tt; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 53 3. if a c > 0, then area of A (A, B, C) = A+P+'t)-* _ P ro o f This follows from the Local Gauss-Bonnet Theorem with 0\ — tt —a, #2 = 7 t — (3, 03 = 7 r — 7 . Thus J ndA = —7r + (ck + j3 + 7 ) . The result follows easily. ■ 3.6 Jacobi fields The Jacobi field can be used as a tool to study the relation between geodesic and curvature, namely, the rate at which geodesics starting from p and tangent to < 7 G TP M spread apart. Intuitively, the rate of spreading of the geodesics is | (dexpp) (w) \ . Given that / (£, s) is a parameterized surface / (f, s ) = expp tv (s) , 0 < t < 1, — e < t < e, (3.40) where p € M, v (s) is a curve in TP M with v (0) = v and v (0) = w, then (dexPP)tv ( M = ( * > °) (3-41) and satisfies the so-called Jacobi equation defined as follows: D efinition 51 Given that 7 : [0, a] — > M is a geodesic in M, then the Jacobi field J is a vector field along 7 satisfying the equation D2 T + R (7 (t), J (*)) 7 (.t) = 0. (3.42) The above equation is called the Jacobi equation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 54 Given that M is a Riemannian manifold of constant sectional curvature k and 7 : [0,2] — > M is a normalized geodesic on M, then the Jacobi equation can be If w (t) is a parallel field along 7 with {7 (t) , w (t)) = 0 and \w (t)| = 1, then The rate of spreading of the geodesics that start at p € M as a function of the sectional curvature k (p, a) is formulated in the following proposition. P roposition 26 Given that p G M, v € TP M, w 6 T„ (TPM) such that |n| = 1, |rr| = 1, (w, v) — 0 and 7 : [0, l\ — > M is a geodesic parameterized by arc length with 7 (0 ) = p, 7 (0 ) = v, then written as (3.43) J (t) — < tw (t), if « = 0 , (3.44) is a solution of the Jacobi equation with initial conditions J (0) = 0 and ^ (0) w (0 ). | J (t)|2 = t2 - (p, 0) t 4 + R(t) (3.45) where K (p, a) is the sectional curvature at p with respect to the plane generated by v and w, and R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 55 In c b d d i't'T /O T b j \J (0 1 “ i ~ \ K ( p > a)t3 + R (0 > (3.46) where Proof. See [Car92, Chap. 5, Prop. 2.7 and Corollary 2.10]. ■ The rays t — > tv (s) deviate from the ray t — > tv (0) with velocity t. In contrast, the geodesic t — > expp (tv (s)) deviates from the geodesic t — » expp (tn (0 )) with a velocity that differs from t by —\K (p, a) t3. 3.7 Hyperbolic spaces with constant curvature The material in this section follows [CFKP97] and [Bea83] closely. In this section, five analytic models for hyperbolic space are introduced. Each model is a complete Riemannian manifold with an associated Riemannian metric. In addition, each model has a constant sectional curvature — 1. The analytic models for hyperbolic spaces are given as follows: 1. Half-space model H n : H n = {(x1,...,xn) e W l :xn >0} (3.47) with the associated Riemannian metric (3.48) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2. Poincare ball model Dn: 56 Dn — {(xi,... ,xn) G M n : x\ H h x2 n < l} (3.49) with the associated Riemannian metric 9 i j (*® 1 ) • ■ • i •^n) 4 Sij (1 — (m ? H h x2 n)) (3.50) 3. Hyperboloid model Hn: EP = {(xx,..., xn+i) G M n+1 : x\ H h x ^ - x 2 n +1 = - l , x n+1 > 0} (3.51) with the associated Riemannian metric Q ij ( * ^ 1 > ■ • • j * ^ n + l ) ~ * 0 i ^ j , 1 i = j ^ n + 1, —1 i — j = n + 1 . (3.52) 4. Jemisphere model J n: Jn = {(xx,..., xn+i) G M n+1 : Xj + • • • + x\ + x£+1 = 1, xn+1 > 0} (3.53) with the associated Riemannian metric Qij ( * ^ 1 j • ■ • j * E n ) Sij X . 71+ 1 1 _ 7*2 _ „ 2 X £ e^2 (3.54) 5. Klein model K n: {(xi,... ,xn) G R" : x\ + • • • + x2 n < 1} (3.55) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 57 w ith the associated Riemannian metric ( \ , X i X j /Q « , ... ,* o = x w + . . . + 4 j + w + W hen n — 2, i72 and D2 can be defined in the complex plane and are referred to as upper half plane and Poincare unit disk models, respectively (see [HenOl, Chap. 7]). It is easily seen that vertical line in the upper half plane are geodesics, from which a Mobius transformation argument yields the general geodesic to be circle orthogonal to the real axis (see [HenOl, Chap. 9]). The same argument holds in the Poincare unit disk. Besides, the geodesic in the disk can be recovered from the geodesics in the upper half plane by another Mobius transformation argument. Given the upper half plane H 2 = {z € C : Im (z) > 0} with Im (2) ’ then the metric pH is given by , . . f \ z - w \ + \z — w\\ pH (z, w) = In :----- zrr—: r j (3.57) ’ 1 \ \ z - w \ + \z - w \J y J and the geodesic for each pair of points z, w in H 2 with z ^ w is the unique Euclidean circle C or line L which contains z, w and is orthogonal to the real axis. The exponential map for this upper half plane model at the point Q.5i is illustrated in Fig 3.1. Given the Poincare unit disk model D 2 — {z G C : \z\ < 1} with the differential 2 \dz\ ds = 1 I |2> 1 — \z\ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 3.1: Image of v = t (cos6, sin9), 0 < t < 1, 6 = 0, . .., under the exponen tial map at 0.5i for the upper half plane. then the metric pD is given by ( \ i |1 — Z\Z%\ + \z 2 — Z i \ . . P d (^1.^ 2 ) — |t ~ z l Z 2 j _ |^2 — 2l| (3'58) and the geodesic for each pair of points z,w in D 2 with z ^ w is the unique Euclidean circle C or line L which contains z,w and is orthogonal to the unit circle. The exponential maps for this Poincare unit disk model is illustrated in Fig 3.2. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 59 0.6 ;- 0.4 ~ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 R e * * ) Figure 3.2: Image of v = t (cos0,sin0), 0 < t < 1, 0 = 0, fg,..., under the exponen tial map at 0.5 (1 + i) for the Poincare unit disk. R eproduced with perm ission o f the copyright owner. Further reproduction prohibited without perm ission. 60 Chapter 4 Coarse Geometry In this chapter, coarse geometry, which is the study of metric spaces from a large scale point of view, is discussed. Coarse geometry is specifically suited for spaces of negative curvature. In fact, a concept of curvature can be defined without being restricted to Riemannian manifolds. This involves the concept of geodesic metric space. This chapter involves the following: First, the geodesic metric spaces; second, the model spaces and CAT (k) spaces; third, hyperbolic metric spaces; finally, quasi isometry (an isometry of coarse geometry) and its properties. The material in this chapter follows [BH99] and [BBI01] closely. 4.1 Geodesic metric spaces D efinition 52 Given that (X, d) is a metric space, then a curve or a path 7 in X is a continuous mapping from a compact interval [a, b ] C l into X. D efinition 53 Given that (X, d) is a metric space, then a geodesic path ^ joining R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 6 1 x £ X to y £ X is a map 7 : [0, l\ — * X such that 7 (0) = 2 7 7 (I) = y, and d (7 (t), 7 (f')) = I* - t'\ , Vt, t' e [ 0 , 1 ]. In addition, a metric space (X, d) is a geodesic space if for every two points in X, there exists a geodesic path joining them with its length equal to the distance between the two points. A metric space (X, d) is a uniquely geodesic space if there is exactly one geodesic path joining two points in X. A metric space (X, d) is an r-geodesic space if for every two points in X of which the distance between them is less than r, there is a geodesic path joining them. A simple example of a geodesic space is the complete metric space (X, d) with the property that Vx, y £ X , there exists a point m £ X such that d (x, m) = d (y, m) = (x, y) (4.1) The Euclidean space En is an example of a uniquely geodesic space. A finite con nected graph can be considered as a geodesic metric space by metrizing the individ ual edges of the graph as bounded intervals of the real line, and then defining the distance between two points to be the infimum of the lengths of the paths joining them, where the length is measured using the chosen metric on the edges. However, an infinite graph need not be a geodesic space; for example, a finite distance graph G that has two vertices and infinitely many edges {en}ff= 1 connecting these two vertices with the length of the edge en equal to 1 + 4 is a complete metric space, but G is not a geodesic space. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 6 2 Definition 54 Given that (X , d) is a metric space, then the length of a curve 7 : [a, b ] — > X is n— 1 H i) = sup Y \d (7 (ti),l(U+i)), a=to<ti<---<tn —b i= 0 where the supremum is taken over all possible partitions (no bound on n) with o - = to < t\ < • • • < tn = b . A curve 7 is rectifiable if it has a finite length. D efinition 55 Given that (X , d) is a metric space, then dis a length metric if \/x, y 6 X, d(x,y ) = inf I (7 ) 7 where the infimum is taken over all rectifiable curves joining x to y. In addition, {X, d) is a length space if dis a length metric. An example of a length space is a Riemannian manifold with the usual distance function. In general, a metric space need not be a length space and a length space need not be a geodesic space. For example, K— {0} with the metric d (x, y) = \x — y\ is a metric space but it is not a length space. In addition, E2 — {0} is a length space but it is not a geodesic space because there does not exist a geodesic joining (1,1)and ( - 1 ,- 1 ) . However, from the Hopf-Rinow theorem, a complete locally compact length space is a geodesic space. T heorem 27 (Hopf-Rinow) Given that X is a complete locally compact length space, then R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 63 1. every closed bounded subset of X is compact, 2. X is a geodesic space. P roof. See [BH99, Part 1, Prop. 3.7]. ■ Hence a complete Riemannian manifold is a geodesic space. Therefore, finitely connected graphs, Cayley graphs, and complete Riemannian manifolds are unified under the concept of geodesic metric spaces. 4.2 Comparison spaces The material in this section follows [BH99, Part 1, Chap. 2] closely. Here some model spaces with constant sectional curvature k are introduced. Later, by comparing the metric properties of a metric space X with the model spaces, the concept of curvature bounds can be defined. The n-dimensional Eu clidean space, the n-dimensional sphere, and the n-dimensional hyperbolic space are introduced as follows: 1. The n-dimensional Euclidean space E" is a vector space M" with a scalar product n {x,y) = Y l xlyl’ (4-2) i—1 where x = (xi,..., xn) and y = (yi, ..., yn) . Given that x ^ y G En, then the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 64 geodesic path joining x to y is (4 3 ) 2. The n-dimensional sphere §" is the set f n + 1 x — (a:1,..., xn+l} £ Rn+1 : (x, x) = (s*)2 — 1 i = i with the metric d : Sn x Sn — > [0,7 r] defined as cosd(x,y) = (x,y), (4.4) where x, y £ Given that x ^ y £ Sn, then the geodesic path joining x to y is the arc of great circle 7 (t) = (cos t) x + (sin t) u, (4.5) where y-(x,y)x is a unit vector with (u, x) = 0 in the (x , y) plane. In addition, the spherical angle between two geodesic paths issued from a point of with initial tangent vectors u and v is the unique number a £ [ 0 , 7r] such that cos a = (u, v ) . 3. The n-dimensional hyperbolic space Hn is the set {x - (xl, .. .,x n+1) £ Rn+1 :{x\x) = —l ,x n+ 1 > 0} R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 65 with the metric d : Hn x 3 B P — » R+, defined such that cosh d (x, y) = — {x | y) , (4.7) where { | ) denotes the inner product defined by {x\y) = (4.8) for all x, y G R"+1. Given that x ^ y G Hn, then the geodesic path joining x to y is where u is the unit vector orthogonal to x (that is (it | u) = 1 , (u | x) — 0 ) in the direction of y + (x \ y) x. In addition, the hyperbolic angle between two geodesic paths issued from a point of H" with initial tangent vectors u and v is the unique number a G [ 0 , 7r] such that cos a = {u\v). Then the model spaces are defined as follows: D efinition 56 Given that k G M , then M ” is a metric space with the following properties: 1. if k = 0, then M f is the n-dimensional Euclidean space E7 1 ; 2. if k > 0 , then M f is obtained from the n-dimensional sphere §r a with distance function such that 7 (t) = (cosh t) x + (sinh t) u, (4.9) cos \fkd {x, y) = (x , y) , where x, y G §"; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3. if k < 0, then M ™ is obtained from the n-dimensional hyperbolic space HP with distance function such that cosh \f^K d (x, y) = — (x | y ) , where x, y € HP. The metric spaces are the standard models which can be described as com plete, simply connected, n-dimensional Riemannian manifolds of constant sectional curvature n. Hence the metric spaces M" are geodesic metric spaces by the Hopf- Rinow theorem (Thm. 27). Proposition 28 Given x £ M", v = ru £ TXM™ , where r is a positive number and u is a vector of unit norm, then in normal coordinates the Riemannian metric ds 2 — Qydx'dxi is given by the formulas £ dr 2 — 4 sinh2 {p/— nr) du2, if k < 0 ; ds2 = dr2 + r 2du2, if k — 0; (4.10) dr 2 + 4 sin2 {s/nr) du2, if k > 0 . Proof. See [BH99, Part 1, Prop. 6.17]. ■ P roposition 29 Given that M ” is a geodesic metric space, then the following as sentations hold: 1. If k < 0, then M ™ is a uniquely geodesic space. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 6 7 Figure 4.1: A geodesic triangle with the corresponding angles. 2. If k > 0, then there exists a unique geodesic segment joining x, y € M ” if and only if d (x , y) < 7r/% /k. Proof. See [BH99, Part 1, Prop. 2.11]. ■ The diameter of M” is denoted by DK. Hence DK = tt/s/k for k > 0 and DK = oofor k < 0 . Definition 57 A geodesic triangle A in a metric space X consists of three points A,B,C £ X as its vertices, and a choice of three geodesic paths [AB], [BC\, [CA\ joining them as its sides. In addition, a geodesic triangle is denoted by A([AB},[BC),[CA}) or A (A ,B ,C ). The following proposition shows the geometrical properties of the geodesic tri angle as shown in Figure 4.1 in Euclidean, spherical, and hyperbolic geometries. Proposition 30 Given a geodesic triangle A (A, B, C) in Mfj with sides of positive length l([BC]) = a, l([CA]) = b, l([AB}) = c and angles a, (3,7 at the vertices R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 6 8 opposite to the sides of length a, b, c, respectively, then the laws of cosine and the summation of internal angles for A (A, B, C) are as follows: 1. If k = 0, then c2 = a2 + b 2 — 2ab cos (7 ), 7r = a + /3 + 7 . 2. If k > 0, then cos (v^c) = cos (y^a) cos (V ^fe) +sin (\/Ka) sin (\A d>) c o s ( 7), 7r < a + /3 + 7 . 5. If k < 0, then cosh (V— «c) = cosh (\/— Kaj cosh (\/— k6 ) — sinh (\/-Ka) sinh (\/— r e f c ) c o s ( 7), 7r > a + f t + 7 . Proof. See [BH99, Part 1, Sec 2.13, The law of cosines in M f ], and the rest follows from Corollary 25. ■ D efinition 58 Given that k is a real number and A (A, B, C) is a geodesic trian gle in a metric space X , then a comparison triangle A (A, B, O') C M 2 for A = A (A, B, C) is such thatd(A,B) = d(A,B), d(B,C) = d (B, C) , d(C,A) = d (C, A ) . A point x € [AB] is a comparison point for x G [AB] if d {A, x) = d (A, xj . Comparison points on [BC] and [CA] are defined in the same way. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 69 Lem m a 31 Given that k is a real number and A (A, B, C) is a geodesic trian gle in a metric space X such that the perimeter of A (A, B, C) , i.e. d(A,B) + d (B, C) + d (C, A) is less than 2DK , then there exists a unique comparison triangle A (A, B, C) C M % for A (A, B, C) up to an isometry of M % . Proof. See [BH99, Part 1, Lemma 2.14]. ■ D efinition 59 Given that n is a real number and A (A, B, C) is a geodesic triangle of perimeter less than 2 DK in a metric space X , then the n-comparison angle between B and C at A the is angle Z ^ (B , C) at A in a comparison triangle A (A, B, C) C M 2KforA(A,B,C). The concept of curvature in Riemannian manifolds can be extended to the more general class of geodesic metric spaces X by comparing each geodesic triangle with its comparison triangle (see [BBI01, Chap. 4 Def. 4.1.9]). The space with upper curvature bounds can be defined by the concept of C AT (k) inequality where CAT stands for Cartan-Alexandrov-Topogonov. D efinition 60 Given that X is a metric space, A is a geodesic triangle in X of perimeter less than 2DK , and A C M ” is a comparison triangle for A , then A satisfies the C AT (k) inequality if for all x, y £ A and all comparison points x ,y € A, d(x,y ) < d(x,y). In addition, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 70 1. if k < 0, X is a geodesic space and every geodesic triangle in X satisfies the CAT (k) inequality, then X is a C AT (re) space; 2. if k > 0, X is a DK -geodesic space and every geodesic triangle in X of perime ter less than 2DK satisfies the C AT (re) inequality, then X is a C AT (re) space; 3. the complete C AT (0) space is called Hadamard space. D efinition 61 A metric space X is of curvature < re, if it is locally a C AT (k) space, i.e. for every x £ X there exists rx > 0 such that the ball B (x,rx) , with the induced metric, is a C AT (re) space. In addition, X is of non-positive curvature, if X is of curvature less than or equal to zero, i.e., d (x, y) < d (x, y) for all x ,y £ A and all comparison points x, y £ A C Mg . Note that the abbreviation C AT («)-space, k £ R, is only used for upper cur vature bounded spaces. In contrast, the spaces of curvatures bounded below by s G l are called the Alexandrov spaces. The spaces of non-negative curvature can be defined in a similar way except that d (x, y) > d (x, y) for all x, y £ A and all comparison points x ,y £ A C Mg . T heorem 32 If X is a CAT (k) space, then it is a CAT (V) space for every k' > n. If X is a CAT (A) space for every k' > k, then it is a C AT (k) space. Proof. See [BH99, Part 2, Chap. 1, Thm. 1.12]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 71 The following theorem shows the equivalence between the curvature and the sectional curvature in a Riemannian manifold. Theorem 33 A Riemannian manifold M is of curvature < k as a geodesic metric space if and only if the sectional curvature of M is < k. Proof. See [BH99. Part 2, Chap 1, Thm. 1A.6]. ■ 4.3 <5-Hyperbolic spaces The material in this section follows [BH99, Part 3.H] closely. Coarse geometry is the study of metric spaces on large scale distances; for example, the lattice Z2 can be seen from far away, as similar to R2. In other words, R2 is a “blurring” of Z2. Conversely, Z2 is a “coarsening” of R2. The 5-hyperbolicity is a version of coarse negative curvature introduced by Gromov. The 5-hyperbolicity can be formulated in several ways. The Gromov hyperbolicity conditions for geodesics spaces are given as follows: D efinition 62 (Slimness) Given that 5s > 0, then a geodesic triangle in a met ric space X is 5s-slim if each of its sides is contained in the union of the 5s- neighborhoods of other two sides. That is, given three points A, B ,C € X and three geodesic segments between them denoted respectively by \BC\ ,[CA], [AB] , then [AB] c { x ' e x \ d{x\ [BC] U [CA]) < 5S} . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 72 Figure 4.2: A 5s-slim geodesic triangle ABC are geodesic triangles Figure 4.3: A tree is (8$ = 0)-slim. The right hand triangle is (Ss — l)-slim (assuming that the length of every link is 1). A geodesic space X is 6s-hyperbolic if every geodesic triangle in X is 8 s-slim. A trivial example of 5-hyperbolic space is a tree which is a O-hyperbolic space. In addition, the n-dimensional hyperbolic space Hn is a 2-hyperbolic space. The following proposition shows that the CAT (k) space, k < 0, is a 5-hyperbolic space. P roposition 34 If k < 0 then every C AT (k) space is 8-hyperbolic, where 8 depends R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 73 only on k. Proof. See [BH99, Part 3.H, Chap. 1, Prop 1.2]. ■ D efinition 63 (G rom ov P ro d u ct) Given that X is a metric space and A e X , then the Gromov product of B, C € X with respect to A is given by the following formula: (B -C)A = l (d (B, A) + d {C, A) — d (B, C) ) . (4.11) £ Definition 64 (4-Point condition) Given that 5G > 0, then a metric space X is da-hyperbolic if (A ■ B )c > min {(A • D)c , (B • D)c} - SG for every A,B,C,D G X. In addition this condition is equivalent to the A-point condition: d (A, B) + d (C, D) < m ax{d (A, C) + d (B, D) , d(A, D) + d(B, C)} + 2SG for all A, B,C,D £ X. D efinition 65 (Insize) Given that A (A, B, C) is a geodesic triangle in a metric space X , and iA £ [BC] , ie € [CA\ , iG € [AB] are such that d (Ia, B) — d(iG, B) = (A ■ C)B x, d(iB,C) = d(iA,C) = (B ■ A)c := y, d(ic,A ) = d(iB,A) = (C ■ B)A ~ z, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 74 v a Za VB c ta Figure 4.4: The internal points of A and the tripod Ta then the tripod T&(x,y,z) is the simplicial metric tree, unique up to isometry, with a central vertex oa and three extremal points vA, vb ,vc such that d(vA,vB) = d (A, B ) , d(vB,vc) = d (B, C), d (vc , vA) = d (C, A ). In addition, the mapping {A,B,C} — * {vA, vB,vc} can be uniquely extended to a map Xa ' ■ A (A B, C) ~ > Ta (% , y, z) whose restriction to each side of A {A, B, C) is an isometry. The x~a ( ° a ) is called the internal points of A and the diameter of Xa ( ° a ) is denoted by insize A . Given that Sj > 0, then a metric space X is 5 1-hyperbolic if every geodesic triangle in X has insizeA bounded by 5j. D efinition 6 6 (Thinness) Given that 5 t > 0, A (A, B, C) is a geodesic triangle in a metric space X , and Xa ' ■ A (A B, C) — > TA (x, y, z) is defined as above, then A (A, B, C) is ST-thin if for every p,q £ Xa W > d (p, q) < 8, for all t £ Ta (a, b, c ). A geodesic space X is 8t-hyperbolic if every geodesic triangle in X is ST-thin. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 75 A Figure 4.5: Fatness of geodesic triangle The concept of d-hyperbolic space can be defined by the fatness of geodesic triangle (see [Roe96, Chap. 9, Sec. Coarse Baum-Connes for hyperbolic spaces ]) as follows: Definition 67 (Fatness) Given that 5p > 0, and A (A, B, C) is a geodesic trian gle in a metric space X, then A {A, B, C) has fatness bounded by 8p, if inf {d ( f x i i y ) T d ( i y , i f ) T d {fz-, i x ) • A £ [SC1 ] , i y G [C54], iz £ [AB]} ^ 8p. A geodesic space X is 8p-hyperbolic if every geodesic triangle in X has fatness bounded by 8p. The following theorem shows the equivalence among the many hyperbolic con ditions. Theorem 35 Given that X is a geodesic metric space, then the following condi tions are equivalent. 1. There exists 8s > 0 such that every geodesic triangles in X is 8$-slim. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 76 2. There exists 8 a > 0 such that the metric space X is Sa-hyperbolic. 3. There exists Si > 0 such that every geodesic triangles in X has insize A < 8j. 4- There exists 5t > 0 such that every geodesic triangles in X is Sr-thin. 5. There exists Sp > 0 such that every geodesic triangle in X has its fatness bounded by 8p. Proof. See [BH99, Part. 3.H, Chap. 1, Prop 1.17 and Prop. 1.20] for (1) through (4). It is obvious that (3) implies (5). The converse can be shown as follows: Given that ix, iy, iz are the points in geodesic triangles where the infimum occurs for the fatness and ic are defined as in Def. 65, then d (ix? iy) T d (iy, iz) T d (iz, ix) T Sp. Then \d (A, iy) d (A, iz)| ^ Sp, \d (B, iz) — d (B, ix)\ < SF, |d(C,ix)-d (C ,iy)\ < Sp. In addition, d(B,ix) + d(C,ix) = d (B, iA) + d (C, iA) , (4.12) d {C, iy) + d (A, iy) — d (C, ip) + d (A, ip) (4.13) d(A,iz) + d(B,iz) = d(A,ic ) + d(B,ic ) ■ (4.14) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 77 Since d (A iB) = d (A, ic) , (4.15) d(B,ic ) = d(B,iA), (4.16) d(C,iA ) = d (C ,is), (4.17) then by adding Eq. 4.12 and Eq. 4.14, and subtracting Eq. 4.13 yields 2 (d (B, ix) - d (B, iA)) = d (C, iy)-d (C, ix)+d (A, iy)-d (A, iz)+d (B, ix)-d (B, iz) Therefore, |d(B,ix) —d(B,iA)\ < ^ 8F. Similarly, \d(C,iy) - d ( C ,i B)\ < \d(A,iz) - d(A,ic)\ < ~SF. Finally, by the triangle inequality, d (ic, Ei) < d(ic ,iz) + d (iz,ix) + d(iA,ix) < 48F. Similarly, d (iA, iB) < 48F d (iB, ic) < 4SF Hence the insize A is bounded by 48F. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 78 4.4 Quasi-Geodesics in Hyperbolic space The material in this section follows [BH99, Part 3.H, Chap. 1 ] closely. Definition 68 Given that (Xi,di) and (X2,d2) are metric spaces, then a (not nec essarily continuous) function f : X \ — > X 2 is a (A, e)-quasi-isometric embedding, if there exist constants A > 1 and e > 0 such that for all x, y € X \ 1 -dx (X, y ) - e < d 2(f (X), / (y)) < Xdi (x, y) + £ If, in addition, there exists a constant C > 0 such that every point of X 2 lies in the C-neighborhood of the image of f, then f is a (A, e)-quasi-isometry and X i and X 2 are quasi-isometric (or coarsely equivalent). Several examples of quasi-isometric spaces are as follows (see [BH99, Part 1. Chap. 8]): 1 . The inclusion Y t-» X from a subset Y of a metric space X is a quasi-isometry if and only if there exist a constant C > 0 such that every point in X lies in the (7-neighborhood of some points of Y. A trivial example of quasi-isometry is the inclusion Z = — » ■ R. In addition, The lattice Z2 is quasi-isometric to the plane M 2. 2. Every finitely generated group F can be considered as a metric space by using the word metric ds associated with a finitely generating set S. If ds' R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 79 is the word metric associated with another finitely generating set S, then the identity map from (r, ds) into (F , dg) is a quasi-isometry. Therefore, every finitely generated group F is a well-defined metric space up to quasi-isometry. 3. Given that a group F acts properly and cocompactly by isometries on a length space X, then F is finitely generated and for any choice of base point x$ € X, the map 7 ^ 7 - j 0is a quasi-isometry from F into X (see [BH99, Part 1, Chap. 8 , Prop. 8.19]). 4. Given that M is a compact Riemannian manifold and M is the universal covering of M with lifted Riemannian metric d, then (m , ctj is quasi-isometric to the fundamental group of M, tti (M ), with the word metric (see [dlH O O , Chap. 4, Cor. 24]). Theorem 36 Given that X and Y are geodesic metric spaces, f : X —> ■ Y is a (A, e)-quasi-isometric embedding, then if Y is 5-hyperbolic then X is S'-hyperbolic where S' depends only on S, A and e. Proof. See [BH99, Part 3.H, Chap. 1, Thm. 1.9]. ■ Definition 69 A (A, e)-quasi-geodesic in a metric space X is a (A, e)-quasi-isometric embedding 7 : I — > X, where I is an interval of the real line (bounded or unbounded) or else the intersection of Z with such an interval. That is, Y \t — t'\ — £ < d (7 (£), 7 (t')) < A \t — t'\ + £ A R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 8 0 for all t, t' € I. I f I = [a, b], then 7 (a) and 7 (b) are called the endpoints 0/ 7 . D efinition 70 Given that X is a metric space, and Ve(A) denotes the e-neighberhood of a set A C X , then the Hausdorjf distance between A ,B C X is dH(A,B) = m f { e \ A c V £(B) and B C V£ (A)} . (4.18) Lem m a 37 Given that X is a 5s-hyperbolic geodesic space and c is a continuous rectifiable path in X , then for every x G 7 , d (x, im (c)) < 53 |log2 l(c)\ + l (4.19) where ^ is a geodesic segment connecting the end points of c. Proof. See [BH99, Part 3.H, Chap. 1, Prop. 1.6]. ■ Lem m a 38 (Tkm ing Quasi-Geodesic) Given any (A, e) quasi-geodesic c : [a, 6] — > X in a geodesic metric space X , then there exists a (A, e' = 2 (A + e))-quasi geodesic d : [a, b]^X with the same endpoints such that: 1. the Hausdorff distance between the images of c and d is less than (A + e) ; 2. for every tx, t2 € [a, b], I {d\[tl,t2]) < kxd{d (tf) ,d (t2)) + k2, where kx = A (A + e) and k2 = (As/ + 3) (A -f- s ) . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 81 Proof. See [BH99, Part 3.H, Chap. 1, Lemma 1.11]. ■ The following theorem shows that quasi-geodesics remain close to the geodesics in hyperbolic spaces. T heorem 39 (Stability of Quasi-Geodesic) For every 5$ > 0, A > 1, e > 0, there exists a constant R = R (5$, A , s) such that if X is a 8s-hyperbolic geodesic space, 7 is a (A, s)-quasi geodesic in X , and [x,y] is a geodesic segment joining the endpoints ofj, then the Hausdorff distance between [x, y) and the image of ^ is less than R. In addition, if e = 0, then the Hausdorff distance between [x,y] and the image of 7 is less than C8$ (1 + log2 A), where C < 100 is some universal constant. Proof. See [BH99, Part 3.H, Chap. 1, Thm. 1.7 ] and [Gro87, Chap. 7, Prop. 7.2.A], ■ Corollary 40 A geodesic metric space X is 5$-hyperbolic for some 83 if and only if, for every A > 1 and e > 0, there exists 8s (depends only on 8s, A , and e) such that every (A,e)-quasi-geodesic triangle in X is 83-slim. Proof. See [BH99, Part 3.H, Chap. 1, Corollary 1.8]. ■ To get a feeling as to how close quasi-geodesics and geodesics are, the up per bounded on the Hausdorff distance between the (A, e — 0)-quasi geodesics and geodesics are derived in both Riemannian geometry and geodesic metric space. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 8 2 4.4.1 Riemannian geometry case Given that 7 : [0, £ ] — ► M is a geodesic and 7 : [0,1 ] — > M is a (A, e = (^-quasi- geodesic with the same end points, via., 7 (0 ) = 7 (0 ), 7 (£) = 7 ^), then the distance from quasi-geodesic to geodesic, i.e., how far the quasi-geodesic can depart from the geodesic, is defined by sup{d(7 (t),7 ) : t € [ 0 , £]}. Suppose that the geodesic is parameterized by the arc length and the quasi-geodesic 7 is parameterized such that for every t e [0,£], d(7 (t),7 ) = d(7(£)>7(t)). Then D(t) := < ^ ( 7 ( ^ ,7 ) = Given that [ 7 (^1),7 (^1)] and [ 7 ^ 2),7 (^2)] are two arbitrarily close geodesics where tx, t2 € [ 0,£], t2 — tx = dt, and q is the point in [ 7 (^2), 7 (^2)] that has minimum distance to 7 (^1), then [ 7 (^1),?] is in the Jacobi field where [ 7 (^2)>7 (^2)] is the nominal geodesic. By the Jacobi field theory, d{l{tx),q) = (t2 - h) cosh(V— ^< ^(7 (^2), q)) = dt cosh(v/—kD (t)) (4.20) since d(7 (ti),7 (^1)) = d(q,'j(t2)). Since the triangle A'y(tx)^f(t2)q coincides at the limit dt — * 0 with its projection exp^t^('y(tx)'y(t2)q) on the tangent space, applica tion of Pythagoras’ theorem in the latter yields d2{l{ti)^{t2)) = dt2 (cosh2(\/~^kD(t)) + (D '(t))2) . (4.21) By the quasi-geodesic property, (cosh2 + (D’(t))2) < A 2. (4.22) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 83 Therefore, the quasi-geodesic 7 hat departs with a maximum speed from the geodesic is given by the differential equation (.D '(t)f - A 2 - cosh2 , (4.23) subject to the boundary condition D( 0) = D{£) = 0 (4.24) and A 2 — cosh2 (\/Ac.D(s)) > 0. (4.25) Therefore, the upper bound between the distance from (A, e = 0)-quasi-geodesic is given by Anax = ~ p = cosh-1 A (4.26) y' — K ~ -L^. In 2A V ~ « Figures 4.6 and 4.7 show the results of two simulation runs of Equation (4.23). In the first case, the quasi-geodesic reaches the bound (4.26) and remains at that constant distance away from the geodesic, while in the second case, the quasi geodesic does not reach its bound. The distance from geodesic to quasi-geodesic, i.e., how far the geodesic can depart from the quasi-geodesic, is defined by sup{d(7 ,7 (t)) : t £ [0,/]} and can be computed by reparameterizing the quasi-geodesic 7 such that for every t € [0,£], d{7,7 (2)) = < 2(7 (2), 7 (£)). Given that 7 is the new parameterization of 7 , R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.6: Envelope of all distance plots between (A = 2,e = 0)-quasi-geodesics and a geodesic of length 20. The curvature k = — 0.05 is adjusted so that the solution reaches the bound, resulting in a continuously differentiable curve. Figure 4.7: Envelope of all distance plots between (A = 2,e = 0)-quasi-geodesics and a geodesic of length 20. The curvature n = — 0.005 is adjusted so that the solution does not reach the bound, resulting in a nondifferentiable curve. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 85 then 7 is still a quasi-geodesic but with a tolerance of A , in general different from A . Given D(t) = 7 (f)), then by the Jacobi field theory and application of Pythagoras’ theorem as in the preceding case, it follows that in the infinitesimal case t2 ~ ~ t\ = dt, = dP (cosh2(V— (£)) + ( ) , where di is the arc length on the quasi-geodesic and D' is the derivative relative to t. By the quasi-geodesic property, (cosh2(\/— K-D(t)) + (D'(t))2) < A 2 From the above, the quasi-geodesic such that the geodesic departs at maximum speed from the quasi-geodesic is easily found and the bound is recovered. That is, the same bound as the previous one, except that A is replaced with A . It remains to find some bound for A . From the definition of quasi-geodesic and arc length, the following inequality can be derived: \-dt < drf < A dt. A Therefore, j t < £(q([0 ,t])) < At and I t < £(*f({0,t})) < It. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 86 Suppose that 7 (t) is between 7 (0) and 7 (i), then it follows that the length of the arc from 7 (5 ) to 7 (5 ) is bounded by (A — A ) t, so that (4.27) d ( 7 - 7) A - i dt A Then d'y < d'y d(7 - 7) < A + A - i dt dt dt A = A . It follow s th a t A = A + a /A 2 - 1. (4.28) (4.29) N ow , su p p o se th a t 7 (f) is b etw een 7 (f) and 7 (£), th e n it follow s th a t the le n g th o f th e arc from 7 (f) to 7 (t) is b o u n d ed b y (A — j ) t and d{l ~ 7 ) dt A - i A d'y < d'y + d{7 - 7 ) < A + X — - dt dt dt X = A . It follow s that A = A + 1 2A ' (4.30) (4.31) (4.32) Given that {A, A + V a2 - 1, A2 + 1 2A } — A + y/ A 2 — 1, (4.33) then the Hausdorff distance between 7 and 7 is bounded as < — F==cosh 1 ( a + a A 2 - 1 j (4.34) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 8 7 Observe that this upper bound does not depend on 5, since in the hyperbolic space with constant curvature the 5 can be easily computed as a function of the curvature. The computation of 5 is shown in Chapter 5. 4.4.2 Geodesic metric space case The upper bound of the Hausdorff distance between the (A, e = 0)-quasi geodesics and geo d esics can b e derived b y u sin g the m a teria l in th e P r o o f o f [BH99, C hap. 3.H.1, T h m . 1.7]. It can b e sh o w n th a t a b o u n d is g iv en b y D max = D q(X2 + 1) + — (2A2 + 3) + A = -D o (A2 + 1) + ^A3 + -A^ (4.35) w here D o is th e m a x im u m so lu tio n to D 0 < 1 + 5 lo g 2 ( ( D 0 (6A 2 + 2) + A (2A2 + 3))) (4.36) and A is a slim n ess o f th e h y p erb o lic m etric space. U sin g so m e eq u a tio n solver, th is b o u n d is easily com p u ted . F ig u re 4.8 show s th e com p arison a m o n g th e tra d itio n a l G rom ov b ou n d (in T h m . 39), th e b o u n d in R iem a n n ia n g eo m etry (Eq. 4.34) an d th e b o u n d in g eo d esic m etric space (Eq. 4.35). C om pared w ith th e tra d itio n a l G rom ov b ou n d , th e refined b o u n d show s a su b sta n tia l im p rovem en t. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. R iefnanniap m anifold bo u n d Geodesic spacfe bound from 0.1 to 1 Lamda from 1 to 2 Figure 4.8: Comparison of Hausdorff distance upper bound: Gromov’ s traditional bound, refined bound in Geodesic metric space, and refined bound in Riemannian manifold. 4.5 k - local Geodesics in Hyperbolic space A n im p ortan t co n cep t in g eo d esic sp a ces is th a t o f a fc-local g eo d esic, defined to b e a con tin u ou s m a p 7 : [a, b ] — » X su ch th a t th e restrictio n 7|[t1,t2] is an iso m etry for every f a — h \ < k. T h e form al d efin itio n for fc-local g eo d esic is g iv en as follows: Definition 71 Given that k > 0 and X is a geodesic metric space, then a path 7 : [a, b \ X is a k-local geodesic if d (7 ( f i ) , 7 f a ) ) = |ti — fa\for all ti, t% G [a, b ] with fa — ^21 < k. The following theorem shows that a fc-local g eo d esic in hyperbolic space is in fact a quasi-geodesic and hence is close to the geodesic. This theorem gives a useful local criterion for constructing quasi-geodesics. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 89 T heorem 41 Given that X is a 5-hyperbolic geodesic space and 7 : [a, 6] — > X is a k-local geodesic, where k > 85, then: 1. The image of'y is contained in the 25-neighborhood of any geodesic segment [7 ( a ) , 7 (6)] connecting its endpoints. 2. [7 (a), 7 (b)} is contained in the 35-neighborhood of the image of 3. 7 is a (A, e)-quasi-geodesic, where e = 25 and A = (k + 45) / (k — 45). Proof. S ee [BH99, P art 3.H, C h ap . 1, T h m . 1.13 ]. ■ A lth o u g h Mocal q u a si-geod esics are close to geo d esics, th e fe-local geod esic can n ot in gen eral b e forced to a g e o d e sic -e v e n if w e ta k e k — » 00, as sh ow n by the following counterexample x: Consider two copies La, Lb of the real line, parallel to each other at a unit distance and such that any common perpendicular crosses th e tw o lin es a t th e sa m e real num ber. G iven th a t th e v ertices are th e integers, d en o ted as ak,bk, k £ Z, on La, Lb, resp ectively, th e n draw th e ed g es [ak, bk], k £ Z. Clearly, this “biinfinite ladder” is Gromov hyperbolic for some finite 5, because it is quasi-isometric to the real line. In addition, the path ao & o & i---& fcaf c is a k-locsl geodesic, but not a geodesic since its length is k 4-2, and the same fact remains true as k — > 00. 1 This counterexample was provided by Prof. Misha Kapovich, University of Utah. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 90 Chapter 5 Computation of metric curvature in Riemannian Geometry In ch ap ter 4, several form u lation s o f th e b a sic m etric h y p erb o lic co n d itio n w ere d iscu ssed . In th is chapter, th e ex p licit form ula for th e variou s < 5 ’s are com p u ted for each geo d esic tria n g le w ith a cu te an gles o n ly in th e h y p erb o lic m o d el sp ace M w ith co n sta n t n eg a tiv e curvature k. T h e S in e an d C osin e ru les for h y p erb o lic sp ace are u sed as d ev elo p in g to o ls for d erivin g th e form ula. T h e form u la for th e 5 d ep en d s o n ly on th e in tern a l an gles o f th e g eo d esic trian gle. In a d d itio n , th e form ula for < 5 reveals th a t th e in fin ite g eo d esic trian gles w ith all v a n ish in g in tern al an gles have th e largest 6 in th e h yp erb olic sp ace M . In ad d ition , th e ’’fa tn ess” o f a geod esic trian gle, defined to b e th e infim um o f th e p erim eters o f all in scrib ed trian gles, can b e giv en a b illiard d y n am ics in terp retation , in th e sen se th a t th e o p tim u m inscribed trian gle is th e p erio d th ree orb it o f th e b illiard d y n a m ics on a g eo d esic triangular ta b le o f co n sta n t n eg a tiv e sectio n a l curvature. T h is in terp reta tio n involves, am on g o th er th in g s, a h y p erb o lic e x ten sio n o f F erm a t’s p rin cip le, in th e sen se th a t th e o p tim a l in scrib ed trian gle ca n b e d efin ed b y its reflection an gles o n th e edges o f th e geod esic tria n g le equal to th e corresp on d in g in cid en ce an gles. T h e o p tim u m R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 91 inscribed triangle is easily constructed as the orthic triangle, that is, the triangle the vertices of which are the feet of the altitudes of the original triangle. Finally, explicit expressions for the orthic triangle, and hence the period 3 orbit, and the fatness are derived in terms of the triangle data. 5.1 Hyperbolic trigonometry With the development of 5-hyperbolic spaces, the hyperbolic conditions (5s-slim, <$T-thin, <5/-insize, and d^-fatness) for geodesic triangles have come to play a cru cial role in the sense that they provide a substitute for the differential concept of curvature that has traditionally been derived from the Riemannian connection. A manifestation of negative curvature of a Riemannian manifold is the fact that it satisfies the 5 hyperbolic conditions. In this section, the hyperbolic measures 5s, St, and 51 are computed for a hyperbolic metric space (M, d) with constant negative curvature k. Given that A, B, C € M with [AB], [B C ], [CA\ the shortest length geodesic arcs joining A to B, B to C, and C to A, respectively, then the geodesic triangles A ABC is [AB] U [BC] U [CA\. This triangle A ABC, shown in Figure 4.1, is uniquely specified up to isometry by the three internal angles a, /?,7 at the vertices A, B, C, respectively, provided that a + + 7 < 7r. Given that A ABC is a geodesic triangle in a hyperbolic space with a sectional curvature k < 0, then the Sine and Cosine Rules for this geodesic triangle A ABC can be summarized as follows: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 92 Lem m a 42 Given that A ABC is a geodesic triangle with three internal angles a, (3, 7 > 0 at the vertices A, B, C, respectively, and a, b, c are the the lengths of the sides opposite to the angles a ,/3,7 , respectively, then the following results hold. 1. The Sine Rule: sinh {yf—~H a) sinh (1 / — «& ) sinh (-</— kc) sin a sin/3 sin 7 2. The Cosine Rule I: cosh (y/^Kaj — cosh fe b ) cosh — sinh (\/— a c & ) sinh (\/— kc) c o s ( a ) , cosh ( \ / — « 6 ) = cosh ( s / — « c ) cosh ( \ / — « a ) — sinh sinh ( V — « a ) c o s ( / ? ) , cosh (\/— f t c ) = cosh (\/— ko) cosh (\/— « & ) — sinh (V— ko) sinh c o s (7 ). 5. T/ie Cosine Rule II: cos /3 cos 7 + cos a cosh (v^ - kg) = cosh (V— «& ) = sin /3 sin 7 cos 7 cos a + cos f3 sin 7 sin a ’ , , ,— \ cos a cos /3 + cos 7 COsh W - K C ) = — ; --------— 5--------• v ' sina sm p Proof. See [Bea83, Chap. 7, Sec. 7.12 ]. ■ In addition, Pythagoras’ theorem for a geodesic triangle with a right angle in hyperbolic space is given as follows: Corollary 43 Given that A A B C is a geodesic triangle with three internal angles, a, f3 < | and 7 = | at the vertices A, B, C, respectively, then the hyperbolic form R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 93 of Pythagoras ’ theorem is given by the following formula: cosh (a / — rec) = cosh ( \/— Ha) cosh (a / — «& ) ■ In addition, the following relations hold: tanh { \f— «5) = sinh (V— fta) tan /3, sinh (a/^k 6) = sinh (\/— kc) sin/3, tanh (y ^ /ta) = tanh ( \/— rec) cos (3 . Proof. See [Bea83, Chap. 7, Sec. 7.11]. ■ The inscribed circle O in a geodesic triangle is a circle that makes contact with each side of the triangle at exactly one point. The following theorem shows how to construct the inscribed circle in a geodesic triangle. T heorem 44 Given that A AB C is a geodesic triangle with three internal angles ck,/?, 7 at the vertices A, B, C, respectively, then the three angle bisectors of A ABC meet at a single point ( in A ABC. This point is the center of the inscribed circle. In addition, the radius R of the inscribed circle of A A B C is given by , , „ cos2 a + cos2 B + cos2 7 + 2 cos a cos B cos 7 — 1 tanh2 R = --------— ------- r-— 1 -------- — .. ■ ■ ■ ....... 2 (1 + cos a) (1 + cos/3) (1 + cos7) Proof. See [Bea83, Chap. 7, Thm. 7.14.2]. ■ Lem m a 45 The area of a hyperbolic circle O whose radius has length R is given by the formula Area (0) = sinh2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 94 where k < 0 is the sectional curvature. Proof. It follows from Equation 4.10 that in normal coordinate, the area of a circle O whose radius has length R is given by f* 2 7 T pR P Z IT p i t / / — ==. sinh (\f^ k r) drdO Jo Jo V~K (cosh (y/~ kR) — l) t v Most of results in section 4.4 and 4.5 are derived for <5s-hyperbolic space. How ever, the computation of the fatness is easier than the slimness. Therefore, the relation between the fatness and the slimness should be established. In case of a Riemannian manifold with constant negative sectional curvature k, this relation can be shown as follows: T heorem 46 Given that M is a Riemannian manifold with negative sectional cur vature k, then every geodesic triangles in M has fatness 8p bounded by and slimness 5s bounded by In addition, 1 < SpSg1 < 6 . Proof. Given that A A B C is a geodesic triangle with three internal angles, a, /3,7 , and the inscribed circle of A A B C has its center at ( and radius R, then the fatness of A AB C is less than 61? by the triangle inequality. Since the area of the hyperbolic triangle = —( . ytg±?) < is greater than the area of the inscribed R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 95 circle = — sinh2 (\/— k§) > irR2, it follows that irR2 < — , so that R < ~r—. — A C \ ® Z / — K > y — /€ Therefore the fatness SF < . It is clear that the slimness of A ABC is less than 2R. Therefore the slimness 5s < In addition, it follows that r < 5$ <2r and 2r < 5F < 6r. Finally, 1 < 5F6gl < 6. ■ The following sections provide the computation of the exact bounds for several metric hyperbolic measures. 5.2 Slimness computation The slimness of the geodesic triangle A A B C is defined as 5s (AABC) := max {5[ab},5[bc],5[ca] \ , where 5[a b \ = sup d(Z, [BC] U [CA]), Ze[AB\ 5 [Bc ] = sup d(X,[CA]U[AB]), xe[B C \ 5[ca] = sup d(Y,[AB]U[BC]). Y£[CA ] Given that X € [BC] and Y € [CA\ , then for each fixed Z 6 [AB], the distances d (Z, X) and d (Z, Y) are given as follows: 1 / sin/5 sinh (xA-k (c — 2)) \ d Z,X) = —; = sinh- I — ------------- ^ ------ 1 ’ 5-1 Y ~ k \ sm 6X j 1 , / sin a sinh ( J — kz) \ i(Z ,Y = — j = sinh- I -------— p ------ I , 6.2) v —k \ sm uv Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 96 where 8 x = Z B X Z and 0y = ZAYZ. Therefore d(Z, [BC]) = m i x e [ B C } d { Z , X ) and d (Z, [CA\) — infyg^] d (Z, Y) occur at the points X , Y where 0 x = &y — f ■ In addition \ZX] is the unique geodesic through Z that is orthogonal to [BC] , and [ZY] is the unique geodesic through Z that is orthogonal to [CA\ . By the intermediate value theorem, there exists a point Z £ [AB] such that d (z,[BC])=d(z, [CA]). (5 .3 ) If Z £ ZB ZA , th en d(Z, [BC] U [CA]) = d{Z, [CA]) < d (z, [CA]) an d if Z £ , th e n d (Z , [BC] U [CA]) = d (Z , [BC]) < d (z, [BC]) . T herefore, 3[ab] = sup d(Z, [BC\U[CA]) Ze[AB] = su p in f { d (Z , [BC]), d (Z, [CA])} Ze[AB] = d (z, [BC]) = d (z, [CA]) . (5.4) G iven th a t z = d (Jz,Al) , th e n w ith d ^Z , [BC]) — d ^Z, [CA]^ , th e follow in g resu lts c a n b e derived: (sin a) sin h \P~kz = (sin (3) sin h % /— ^ (c — z), S m ^ sin h ( \ / —k 5) = sin h ( \ / —« c) cosh (V— — cosh (V ~ -« c ) sin h (\/— , c o th (\/— = ..... ....f c o s h ( v ^ c ) + , sin h ( C —Kc) \ s m /3/ 1 / , / /— \ sine 1 " , c = --------9 , _ _ > ------ ( cosh ( \ / — k c ) + — sinh (y — k z ) j cosh ( y — «c) — 1 \ si sin/3 2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 97 The Cosine Rule II in A A B C yields ✓ \ 2 2 1 \ (sina sin/3) 1 + . sinh ( y / —kz) J (cos a co s (3 + cos 7 )2 — (sin a sin /3 2 co s a cos /d + cos 7 ^ sin a sin h 2 (\/— = ( (sin a ) sin h y / —n z ) 2 = T herefore, sin a sin /3 sin/d, (cos 7 + cos a cos (3 + 1 — cos2 a)2 (2 cos a cos /3 cos 7 + cos2 a + cos2 j3 + cos2 7 — 1) ’ (2 cos a cos (3 cos 7 + cos2 a + cos2 (3 + cos2 7 — 1) (2 + 2 cos 7 — (cos a — cos /3)2) (sin2 a) (2 cos a cos f3 cos 7 + cos2 a + cos2 f3 + cos2 7 — 1) (2 + 2 cos 7 — (cos a — cos y d )2) 1 . , 1 / (2 cos a cos (3 cos 7 + cos2 a + cos2 (3 + cos2 7 — 1) 7 = S inh J -----------------; .................. ...........;--------------------- —9 7--- — ----- — . — ft y (2 + 2 cos 7 — (cos a — cos p) ) Sim ilarly, . 1 . , _i / (2 co s a co s /3 cos 7 + co s2 a + co s2 {3 + co s2 7 — 1) J (2 + 2cosa — (c o + — cos'^7)------------ (5'6) ft and 1 , , / (2 cos a co s y d cos 7 + co s2 a + cos2 /3 + co s2 7 — 1) = sm h , / -------------------------------- ; ----------------------------------------------------------------------------- - 7 7 ------- ( 5 .7) V (2 + 2 cos y d — (cos 7 — co s a ) ) &[CA\ ~ Finally, it is conjecture that the maximum slimness triangle is obtained when a = /3 = 7 = 0, that is, the ideal triangle with its vertices at infinity. Therefore the slimness of the Riemannian manifold with constant curvature ft Is given by 8 s = - 7 == sinh-1 (1) « 1 . —0.8814. (5.8) -ft v — ft R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 98 5.3 Insize com putation The insize of the geodesic triangle A ABC is defined as Si (AABC) := sup {d (iA, iB) , d (iB, ic ) , d (ic , iA)} , where iA e [BC] ,iB £ [CA], ic 6 [AS] are such that d(A ,B ) = d (ic ,S ) = (A-C)B ^ C + ^ ~ 6, = d (u , C) = (B • A)c = ~-C, d(ic,A) - d(iB,A) = (C-S)A = - - ± | - - . The Cosine Law in A icB iA yields cosh [yfA^d (ic, iA)) = cosh (y^dcd (iA ) S)) cosh (\/— (icj B)) — sinh (a/— (A, B)) sinh {\/—nd (ic, B)) cos /? = cosh2 (^J— nd (iA, B)) — (cosh2 ( \/— ^d (iA, B)) — l) cos ft = (1 — cos /3) cosh2 (-</— Kd (u , B)) + cos fd. (5.9) Since 2d(iA, B) = c + a — b, it follows that cosh (2\/— nd (iA, B)) = cosh (V— « (c + a — 6)) . (5.10) The left-hand side of Eq. 5.10 yields cosh (2'yJ—nd (iA, B)) = cosh2 ( \A d d (iA, B)) + sinh2 ( \/— ~itd (A, B)) — 2 cosh2 (s/A^d(iA,B)) — 1. (5.11) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The right-hand side of Eq. 5.10 yields cosh (a/ — « (c + a cosh (\/— cosh (V— k6) cosh (a/^kc) — c o s h (^/— k c l) sinh ( a/ — «& ) sinh ( a/ — k c) + cosh (\J— K ,b ) sinh (V— kc) sinh (>/— rea) — cosh (V— kc) sinh (a/— sinh (V— k& ) . (5.12) From the Cosine Rule I, the following expressions can be derived: sin h (a/ —k&) sin h (a /^ k c ) sinh K /—kc) sinh ( a J—koS sinh (y/—Ka) sinh (\/— «$) cosh ( a/ —k6) cosh (\Z— kc) — cosh (yf^KO) cos a c o s h ( a/ — Kc) c o s h (y/~K(l) — COSh (y/~K,S) cos (3 cosh (V— ko) cosh (y/—K ,$ j — cosh (y/— kc) cosy , (5.13) , (5.14) . (5.15) Substituting Eq. 5.13 through 5.15 into Eq. 5.12 yields the following: cosh (V— k (c + a — b)) cosh ( a/ — Tea) cosh (\/— a& ) cosh (a/^kc) cos a cos /3 cos 7 • (cos a cos 7 — cos a cos /3 — cos j3 cos 7 + cos a cos /3 cos 7 ) 1 cos a cos (3 cos 7 (cosh2 (y/— naj cos { 3 cos 7 — cosh2 (\/— « & ) cos a cos 7 + cosh2 ( -kc cos a cos 0) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 100 The Cosine Rule II in A ABC yields cosh (v~-ko) = cosh [ y / —~Kb) = cosh (•>/—kc) = H ence co sh ( 1 /—« (c + a — &)) 1 \ ( 1 cos /3 cos 7 + cos a sin (3 sin 7 ’ cos 7 cos a + cos /3 sin 7 sin a ’ cos a cos /3 + cos 7 sin a sin f3 cos a cos /3 cos 7 / \ sin a sin /? sin 7 y (cos a cos 7 — cos a cos /3 — cos f3 cos 7 + cos a cos j3 cos 7 ) (cos f3 co s 7 + cos a ) (cos 7 cos a + cos /3) (cos a cos j3 + cos 7 ) 1 x 2 cos a c o s /? cos 7 ) \ sin a sin f3 sin 7 y (cos /3 cos 7 + cos a ) 2 ( l — co s2 a ) cos (3 cos 7 1 ^ cos a cos P cos 7 / \ sin a sin f3 sin 7 , (co s 7 cos 0; + cos f3)2 ( l — co s2 /3) cos a cos 7 1 + cos a cos /3 cos 7 / \ sin a sin /3 sin 7 , (co s a cos (3 + cos 7 ) 2 ( l — co s2 7 ) cos a cos f3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 101 2 cosh2 {^f^Kd (iA, £»)) — 1 = cosh (\f—n (c + a — b)) ( 1 \ 2 = — ------:——— ) (1 — cos a) (1 + cos/3) (1 — cos 7) \ s 1 nasmpsm 7 ) ^ cos a— cos (5+ cos 7 — cos a cos f 3+ cos a cos 7 — cos /3 cos 7 +cos a cos fd cos 7 + cos2 a+cos2 jd+cos2 7 cosh2(V ^ d { iA,B)) = l + cos7 + A (1_ cosa)(1+cos^)(1_ COS7) v 2 \ smasmpsm 7 J Finally, cosh ( \/— ~K d (ic, iA)) = (1 — cos /?) cosh2 (v^— /cd (iA, B)) + cos fd 1 (cos a — cos /? + cos 7 + l ) 2 — ~ 77---------------- T77----------------7 ---------h C O S p 2 (1 + cos a) (1 + cos 7 ) _ (2 cos a + 2 cos 7 + 2 cos a cos 7 + 2 cos a cos jd cos 7 + cos2 a+cos2 jd + cos2 7 + 1) 2 ( 1 + cos a) (1 + cos 7 ) ’ d (ic, ijC j = — p=cosh_1 Y — K (2 cos a + 2 cos 7 + 2 cos a cos 7 + 2 cos a cos j3 cos 7 + cos2 a+cos2 /3+cos2 7 +1) 2 (1 + cos a) (1 + cos 7 ) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 102 Similarly, d(iA,iB) = 3 cosh-1 ■S/— K (2 cos a+2 cos j3-f 2 cos a cos/3+2 cos a cos /3 cos 7 + cos2 a+cos2 /3+cos2 7 + 1) 2 (1 + cos a) (1 + cos/3) (2 cos /3+2 cos 7 + 2 cos /3 cos 7 + 2 cos a cos j3 cos 7 + cos2 ai+cos2 /3+cos2 7 +I) 2(1 + cos /3) (1 + cos 7 ) Finally, it is conjecture that the maximum insize triangle is obtained when a — /3 — 7 = 0. Therefore the slimness of the Riemannian manifold with constant curvature k is given by Sj = - t L : cosh-1 (1.5) « -^=0.9624. (5.16) V —K , \/—K d (iBAc) — r —cosh 1 V—K 5.4 Thinness computation The thinness of the geodesic triangle A ABC is defined as 5 t (AABC) = sup {SA, SB, < 5 C} , where 5a — sup {d (v, w) : v £ [iBA], w £ [icA], and d (v, A) = d (w, A)}, SB = sup {d (w, u) : w £ [icB], u £ [iAB \, and d (w, B) — d (u, B)} , 8c = sup {d (u, v) : u £ [iAB], v £ [icB], and d (u, C) = d (v, C)} . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 103 Given that w € [icB], u € [% a B ] , and d (w, B) = d (it, 5 ) , then the Cosine law in AwBu yields cosh {yf^Kd (w,u)) — cosh ( \/— B)) cosh [y/^nd (u, Bj) — sinh [^f—nd (w, £?)) sinh [\/—Kd (u, B)J cos /3 — (1 — cos /?) cosh2 (\f^ n d (it, 5 )) + cos /3 < (1 — cos /3) cosh2 (\f^ n d (iA, 5 )) + cos /? Therefore, Similarly, < 5 # = d(ic,lA) Sa = d(iB,ic ) < 5 c = d(iA,iB) Hence < 5 r (AABC) = < S 7 (AAJ5C). (5.17) Therefore the slimness of the Riemannian manifold with constant curvature k is given by St = r— cosh 1 (1.5) ~ r— 0.9624. K K (5.18) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 104 5.5 Fatness com putation The fatness of a geodesic triangle is defined to be the infimum of the perimeters of all inscribed triangles, that is, the fatness of the geodesic triangle AABC is defined as follows: SF {AABC) := M{d(X,Y)+d(Y,Z)+d{Z,X):X e[BC],Y e[CA],Z e[AB]}. The fatness of a geodesic triangle is given a billiard dynamics interpretation, in the sense that the optimum inscribed triangle is the period three orbit of the billiard dynamics on a geodesic triangular table of constant negative sectional curvature. This involves, among other things, a hyperbolic extension of Fermat’s principle, in the sense that the optimal inscribed triangle can be defined by its reflection angles on the edges of the geodesic triangle equal to the corresponding incidence angles. The optimum inscribed triangle is easily constructed as the orthic triangle, that is, the triangle the vertices of which are the feet of the altitudes of the original triangle. The explicit expressions for the orthic triangle, and hence the period 3 orbit and the fatness, are derived in terms of the triangle internal angles a, 13 , 7 in the next section. The fatness of the geodesic triangle AABC is given by the following formula: Sp (AABC) = * - j = . sinh ^2-\/(2cosacos/3cos7+cos2a+cos2/3-fuos27)(2costtcos/3cos7HcQs2a+cos2/?+cos2 7— ljj R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 105 5.6 Billiard dynamics interpretation of fatness of hyper bolic geodesic triangles In this section, the explicit formula, rather than a bound for the fatness of an ar bitrary geodesic triangle with acute angles only on a constant negative curvature Riemannian manifold, is derived. It turns out that the fatness corresponds to the (unique) period 3 orbit of the billiard dynamics on a hyperbolic geodesic triangular table. An outline of this section follows: First, the Euclidean triangular billiard table, which does have a periodic orbit of period 3, referred to as Fagnano periodic orbit, is reviewed. In fact, this orbit is easily constructed as the orthic triangle from the altitudes of the triangle (see Figure 5.1). To extend this result to hyperbolic ge ometry. the hyperbolic Fermat principle, saying that under the first order optimally conditions the smallest perimeter inscribed triangle has its incidence angles equal to the reflection angles, is derived. In fact, this result can be proved, synthetically, by a hyperbolic extension of Fejer’s construction. However, this does not provide analytical expressions of the angles involved. An alternative proof involving some hyperbolic trigonometry manipulations is presented here. This proof provides the expression of the incidence and reflection angles in term of internal angles. Next, the existence and uniqueness in hyperbolic space of an inscribed triangle with its incidence angles equal to its reflection angles is proved. Then, the second order variation is derived. Finally, the fatness formula is derived. The obtuse angle case R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 106 A orthocenter altitude altitude X altitude Figure 5.1: The Fagnano period orbit, shown to correspond to the orthic triangle. is dealt with at the end of this section, because for the reason explained there, it does not have a billiard dynamic interpretation. 5.6.1 Euclidean triangular billiard table Given that A A B C is a geodesic (rectilinear) triangle without obtuse angles in E2, finding the minimum perimeter triangle inscribed to A A B C is the celebrated Fagnano problem [Cox89], which has the following solution: From A, draw the altitude A X , that is, the line segment such that X G [BC] and [AX] ± [BC]. Likewise, draw the altitudes [BY] and [CZ], As is well known, the three altitudes intersect at a single point, referred to as orthocenter H. It turns out that A X Y Z , referred to as orthic triangle [Cox89], is the period 3 orbit of the billiard dynamics in the sense that ZYXC — ZZXB with the same fact at the points Y, Z. The traditional Fermat principle of geometrical optics [YM68] is enough to prove that R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 107 the period 3 orbit [XY] U [YZ] U [ZX\ of the billiard dynamics is the minimum perimeter inscribed triangle. The proof that the orthic triangle yields a period three orbit is very simple; however, it is necessary to identify the key Euclidean geometry argument leading to this result, with the objective of assessing whether it can be extended to constant negative curvature geometry, also referred to as Lobachevskian geometry [Smo82]. Among these proofs, the Fejer construction [Cox89, Sec. 1.8] utilizes an argument based on the reflection of [AX] across the edges [AB] and [AC], along with the orthogonality properties of the orthic triangle, to prove directly without appealing to Fermat’s principle that the orthic triangle is the minimum perimeter inscribed triangle. The Fejdr construction is given as follows (see Figure 5.2): Given that X < E [BC] is fixed, then find Z e [BA], Y € [AC] such that d(X, Y) + d(Y, Z) + d(Z, X) is minimized. Reflect X across [AB] to get X] likewise, reflect X across [AC] to get 1 . Clearly, d(X, Y) + d(Y, Z) + d{Z, X) = d{X, Z) + d{Z, Y) + d{Y, X), so that at optimality, X, Z, Y, X are aligned, which implies that Z B Z X = ZAZY and ZAYZ = Z.CYX, that is, the Fermat Principle. To find the optimum X, observe that the angle at A of the isosceles triangle X A X is twice Z.BAC, so that it does not depend on X. Clearly then d(X, X) is minimized iff d(X, A) = d(A, X) — d(A, X) is minimized, that is, [AX] ± [BC]. A similar arguments holds true for the other points. Hence the orthic triangle yields the minimum perimeter inscribed triangle. This argument can be extend easily to the hyperbolic case by substituting the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 5.2: The Fejdr construction, showing that the orthic triangle is the periodic three orbit. The construction remains valid in hyperbolic geometry. hyperbolic geometry concept of inversion (which is conformal and hence preserves the angles [Smo82, Thm. 9]) for the Euclidean concept of reflection. 5.6.2 First order conditions Given that AABC is a geodesic triangle in a constant negative curvature hyperbolic space M, then this triangle is uniquely specified up to isometry by the three internal angles a, j3 , 7 at the vertices A, B, C, respectively, provided that a+f3-\-/y < 1 r. Given that a,b,c are the lengths of the sides opposite to the angles a, f t , 7, respectively, and X,Y, Z are arbitrary points in [BC], [CA], [AB], respectively, then X can be defined as a mapping which maps x £ [0, a] to the point X (x) £ [BC], such that d (X (x) ,B) = x , with similar definitions for Y and Z. Given that F (x, y,z) — d (X(x),Y(y)) + d (Y(y), Z{y)) + d (Z(z),X(x)), R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 109 5f(AABC) = inf < it follows that / 0 < x < a F (x, y, z) : o < y < h 0 < z < c To simplify the computation in the following section, the curvature is assumed to be equal to — 1. For arbitrary curvature k < 0, the following results will become applicable after multiplying the distance inside the argument of the hyperbolic trigonometry functions by \/—k. Clearly, the fatness is to be computed via the hyperbolic Cosine Rule I: cosh (d(x,y)) = cosh ((a — x)) cosh (y) — sinh (a — x) sinh (y) cos (7 ), cosh (d (y,z)) — cosh ((6 — y)) cosh (z) — sinh (b — y) sinh (z) cos ( a ) , cosh (d (z, x)) = cosh ((c — z)) cosh (x) — sinh (c — z) sinh (x) cos (/3), where d(x, y) is a short for d(X(x),Y(y)) with a similar convention for d(y, z) and d(z,x). Taking partial derivatives of the above expressions yields the following results: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 110 ~ cosh (d(x,y)) — sinh (d (x, y)) -— -d (x, y) ( J X ( J X Q — (cosh (a — x) cosh (y) — sinh (a — x) sinh (y) cos (7 )) (JX — sinh (a — x) cosh y + cosh (a — x) sinh y cos 7 , d d — cosh (d (z, x)) = sinh (d(z,x))— d(z,x) \J X ( / X d = — (cosh (c — z) cosh (x) — sinh (c — z) sinh (x) cos (/?)) OX — cosh (c — z) sinh x — sinh (c — z) cosh x cos jd. Manipulating the above, the explicit expressions for the first order partial deriva tives in terms of the triangle data are as follows: d , . _ sinh (— a + x) cosh y + cosh (— a + x) sinh y cos 7 dx X' ^ sinh (d (x, y)) ’ d . . cosh (— a + x) sinh y + sinh (— a + x) cosh y cos 7 dy sinh (d (x, y)) ’ d — d(x,y) = 0 , d . , sinh (—h + y) cosh z + cosh (—b + y) sinh z cos o l dy sinh (d (y, z)) ’ c ? , . cosh ( —6 -f y) sinh z -f sinh ( —6 + y) cosh z cos a dz ^’Z sinh (d (y, z)) ’ ■^d (y,z) = 0 , d ,. . sinh (— c + M cosh x + sinh (— c + z) sinh x cos B — d(z,x ) = ......... , dz sinh (d (z, x)) d ,, . cosh (c — z) sinh x — sinh (c — z) cosh x cos B — d (z, x) = ........ -.......... W w dx smh (d (z,x)) d — d (z, x) = 0 . dy Cancelling the first order variation of F (x, y , 0) relative to x yields the following R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. I l l result: which implies that 3 3 — F (x, y, z) = — (d (x, y) + d {y, z) + d (z, x)) 3 — (d(x,y) + d(z,x)) 3 3 — d(x,y) = - — d(z,x). (5.19) The Cosine Rule I in AY X C yields cosh (d (x, y)) cosh (a — x) — cosh (y) cos (ZYXC) = sinh (d (x,y)) sinh (a — x) (cosh(a— x)cosh(y) — sinh(a— x)sinh(y)cos (7 ))cosh(a — x)—cosh(y) sinh (d (x, y)) sinh (a — x) cosh2 (a— x)cosh(y)—sinh(a— ®)sinh(j/)cos (7 )cosh(a— x) — cosh(y) sinh (d (x , y)) sinh (a — x) sinh2 (a — x) cosh (y) — sinh (a — x) sinh (y) cos (7 ) cosh (a — x) sinh (d (x, y)) sinh (a — x) sinh (a — x) cosh (y) — sinh (y) cos (7 ) cosh (a — x) sinh (d (x,y)) r \ - — d(x,y) (5.20) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 112 The same Cosine Rule I in AZX B yields , . _ cosh (d (z, x)) cosh (x) — cosh (c — z) sinh (d (z, x)) sinh (x) _ (cosh(c— ^)cosh(rr) — sinh(c— z)sinh(x)cos (/3))cosh (x) — cosh(c—z) sinh {d (z, x)) sinh (x) cosh(c— z)cosh2(x) — sinh(c— z)sinh(x)cos (/3)cosh(x) — cosh(c— z) sinh (d (z, x)) sinh (x ) _ _ cosh (c — z) sinh2 (x) — sinh (c — z) sinh (x) cos (0 ) cosh (x ) sinh (d (z, x)) sinh (x) _ cosh (c — z) sinh (x) — sinh (c — z) cos (/3) cosh (x) sinh (d (z, x)) r \ = fa.d (z'x) (5-21) Combining Eq. 5.19, 5.20, and 5.21 yields cos (ZYXC) = cos (Z Z X B ) =: cos (0X ) (5.22) This is the hyperbolic Fermat principle, saying that a light ray emanating from Y, reflecting at X G [BC], to reach Z would have its reflection angle equal to its incidence angle. Next, cancelling the first order variation relative to y, z yields the following results: cos (ZZYA) = cos (ZXYC) =: cos (0y) (5.23) cos (ZXZB) = cos{ZYZA)=:cos(9z). (5.24) For the optimization problem to be a differentiable one, it is hence necessary that there exists an inscribed geodesic triangle AX Y Z such that the reflection angles of its edges on A ABC equal the corresponding incidence angles. In Euclidean R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 113 geometry, this is equivalent to saying that A ABC has acute angles only. The argument in hyperbolic geometry is, however, more complicated. 5.6.3 Hyperbolic orthocenter construction It is easily seen that, for a hyperbolic geodesic triangle A ABC, there exists a point X G [BC ] such that [AX] A [BC] if the angles ZABX and ZACX are acute. Therefore, if the triangle A ABC has no obtuse angles, there are points X G [BC], Y G [AC], Z G [AB] such that [AX] 1 [BC], [BY] A [AC], [CZ] A [AB], respectively. Even though it is not known at present whether [AX] fl [BY] fl [CZ] ^ 0 , this construction yields an inscribed triangle AXYZ, which, as proved in this section, has the property that its reflection angles on the edges of A ABC equal the corresponding incidence angles. Hyperbolic trigonometry of the right-angled subtriangles of A ABC yields tanhx = tanhccos/3, tanh y = tanh a cos 7 , tanh z = tanh b cos a , tanh (a — x) = tanh b cos 7 , tanh (b — y) = tanh c cos a, tanh (c — z) = tanh a cos /3. From the Cosine Rule I applied to the triangles A Z B X and A Y C X and Pythago R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 114 ras’ theorem, d (z, x) and d (x, y) can be expressed as follows: cosh d(z,x) = (cosh (c — z) cosh x — sinh (c — z) sinh x cos j3) — (cosh (c — z) cosh x) (1 — tanh (c — z) tanh x cos /3) = (cosh (c — z) cosh x) (l — tanh c tanh a cos3 /?) cosh d(x,y) = (cosh (a — x) cosh y — sinh (a — x) sinh y cos 7 ) = (cosh (a — x) cosh y) (1 — tanh (a — x) tanh y cos 7 ) = (cosh (a — x) cosh y) (l — tanh a tanh b cos3 7 ) Given that 9X denotes ZZXB and 6r x denotes Z.YXC, then the Sine Rule in the triangles AZ B X and A Y C X yields the following results: • 2 m /■ 2 a\ sinh2 (c — z) t ■ 2 o\ sinh2 (c ~ z) sm = (sm ^ a H m = (sm 0) cosh = (cin2/3) ------------------------------------------ {6 26) (cosh (c—z) coshx) (1 — tanh a tanh c cos3 (5) — 1 • 2 nr / . 2 \ sinh2 (y) /. 2 \ sinh2 (y) sm2C = (1 sm2 7 ) = (sin 7 ) sinh d (x, y) cosh d (x, y) — 1 = (sin2 7 ) ------------------------------------ (5.26) (cosh (a—x) coshy) (1— tanh a tanh b cos3 7) —1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 115 Now, observe the following computation: sinh2 (c — z) = ^ 1 — tanh (c — z) 1 — (tanh a cos p) cosh2 (c — z) = ------ - ^ 7 r — ----------- 772 (5-28) 1 — tanh (c — z) 1 — (tanh a cos/3) . , o tanh2 y (tanh a cos 7 I2 . sinh2 y = ---------- V = — ---------------” 9 (5.29) 1 — tanh y 1 — (tanh a cos 7 ) cosh2 y = - 9 = -7; (5.30) 1 — tanh y 1 — (tanh a cos 7 ) cosh2 x — ........... 9 = ------------ 1— _ —_ (5.31) 1 — tanh x 1 — (tanh c cos 0 ) cosh2 (a — x) — ---------- ---------- - —----------------------k (5.32) 1 — tanh (a — x) 1 — (tanh b cos 7 ) Substituting the expressions in Eq 5.27 through 5.32 into Eq. 5.25 and 5.26 yields the following results: sin2 el x = (sin2 0 ) (tanh a cos ff)2 1— (tanh a cos /3)2 ( r~7T~r a , U (1 — tanh a tanh c cos3 /?)2 — 1 \ 1 — (tan h a cos /3) 1— (tanh c cos 0) J ' ' (sin2 0 ) (tanh a ) 2 (l — (tanh c cos/?)2) (tanh2a — 2tanh a tanh c cos /3+tanh2c— tanh2a tanh2c cos2/? +tanh2a tanh2c cos4 /?)’ ,2 nr (tan h a cos 7)2 sin2 dr x (sin2 7 ) 1 — (tanh a cos 7) ---- 1 ? 7 -77- r -------- ) (1 - tanh a tanh 6 cos3 7)2 - 1 1 — (ta n h6cos7) 1— (ta n h a c o s7) y ' / / (sin2 7) (tanh a)2 (l — (tanh b cos q)2) (tanh2a— 2tanh a tanh b cos7-f-tanh25— tanh2a tanh25 cos274-tanh2a tanh26 cos4 7 ) With the application of Cosine Rule II, tanh a, tanh 6, and tanhc can be written R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 116 as follows: , 2 cosh2 a — 1 tanh a — — — 9---- cosh a (cos 0 cos 7 + cos a )2 — (1 — cos2 (3) (1 — cos2 7 ) (cos 0 cos 7 + cos a)2 _ (2 cos a cos /3 cos 7 + cos2 a + cos2 /3 + cos2 7 — 1) (cos (3 cos 7 + cos a)2 , o , (2 cos a cos (3 cos 7 + cos2 a + cos2 (3 + cos2 7 — 1) tanh 0 = ------------------ ■ ----■ ------- ■ ----■ —— - j------------------ (cos 7 cos a + cos f3) , , (2 cos a cos 0 cos 7 + co s2 a + co s2 8 + co s2 7 — 1) tanh2 c = -----------------------------------------j — ..........— — (5.35) (cos a cos f3 4- cos 7 ) Finally substituting tanh a, tanh b, tanh c by their values into the expressions of sin2 0l x and sin2 9r x yields, (5.33) (5.34) . 2 M ■ 2 nr cos2 j3 + cos2 7 + 2 cos a COS p cos7 /r sm 9X = sm 6X = — ----------— ------- -------------------- ------- (5.36) cos a + cos^ p + cos 7 + 2 cos a cos p cos 7 This proves that the reflection angle at X equals the incidence angle at the same point and can be written as follows: % = ( J cos2 y d + cos2 7 + 2 cos a. cos j3 cos 7 \ ■ ■ ' 1 (5.37) y cos2 a + cos2 0 + cos2 7 + 2 cos a cos 0 cos 7 I The same fact is easily proved for the points Y, Z. Therefore, the hyperbolic orthocenter construction yields an inscribed triangle with its incidence angles equal the corresponding reflection angles. Since it follows from the preceding that the altitudes [AX], [BY], [CZ] of the triangle A ABC are the angle bisectors of the triangle AXYZ, then the following result emerges: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 117 C orollary 47 The altitudes of a constant curvature hyperbolic geodesic triangle with acute angles only intersect at a single point, called hyperbolic orthocenter. Proof. The altitudes [AX], [BY], [CZ] of the triangle A ABC are the angle bisectors of the triangle AX Y Z and hence intersect in a single point by Thm. 44. ■ The above will fail in case the triangle has an obtuse angle. To be more specific, in the Poincare disk model, the altitudes of a triangle with obtuse angles might not intersect in the disk. 5.6.4 U niqueness In this section, the uniqueness of the inscribed triangle that satisfies the first order variation conditions is proved. That is, the only inscribed triangle that has its incidence angles equal to the corresponding reflection angles at X, Y, Z is the orthic triangle. Given that A X Y Z is an inscribed triangle which incidence angles equal to the corresponding reflection angles at X, Y, Z, and denoting these angles by 6x,9y,9z, respectively, then the Cosine Rule II for AX C Y yields cosh d (X, C) cos 9X cos 7 + cos 9y (5.38) cosh d (Y, C) sin 9X sin 7 cos 9y cos 7 + cos 9X (5 .3 9 ) sin 9y sin 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 118 which implies that sinh d (X, C) = JcoslE d (X, C) - 1 a / (2 cos7 cos 0X cos9y + cos2 7 + cos2 8X + cos2 6y — 1) fc. A n A - - - ,(5.40) sm ux sm 7 sinh d (Y C) - a/ ( 2 cos 7 C Q s cos 0y + cos2 7 + cos2 9X + cos2 0y - 1) ’ sin sin 7 By the application of the Cosine Rule II for A ABC, a similar computation yields the following formula: . , , . cos a cos 6 cos 7 + cos2 a + cos2 8 + cos2 7 — 1 * sinh (a) = v ........................ .: .rfis.• / 'S ...............................» (5- 42) sm (p) sm (7 ) . , a / 2 cos a cos B cos 7 + cos2 a + cos2 8 + cos2 7 — 1 sinh (b) = 7 ' ..../-v..........-................... , (5.43 sm (7 ) sm (a) . , , . a/ 2 cos a cos/3 cos 7 + cos2 a + cos2 B + cos2 7 — 1 . , sinh(c) = ...................................... ------------------------ • (5.44) sm (a) sm (p) By using the expressions in Eq. 5.38 through 5.44, cosh (d (X, B )) and cosh (d (Y, A)) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 119 can be expressed as follows: cosh (d (X , B)) — cosh (a — d (X, C)) = cosh (a) cosh (d (X, C)) — sinh (a) sinh (d (X, C)) _ cos (/3) cos (7 ) + cos (a) cos (7 ) cos (Bx) + cos (0y) sin ((3 ) sin (7 ) sin (7 ) sin (6X ) ■sj2 cos a cos P co s 7 + co s2 a + co s2 (3+ co s2 7 —1 sin (/3 ) sin (7 ) y / 2 cos 7 cos 9X cos 6y+ cos2 7 + cos2 0X+cos2 9y — 1 ^ ^ _ _ _ _ _ _ _ _ , (5-4 5) cosh(d(F,A)) = cosh (h — d (Y, C)) = cosh (6) cosh (d (Y, C)) — sinh (6) sinh (d (Y, C)) _ _ cos (7 ) cos (a) + cos (/3) cos (7 ) cos (0y) + cos (0^) sin (7 ) sin (a) sin (7 ) sin (0y) \J*l cos a cos (3 cos 7 + cos2 ct+cos2 /3+cos2 7 — 1 sin (7 ) sin (a) J 2 cos 7 cos 9X cos 0y+ cos2 j + cos2 6X + cos2 8y — 1 ------------------------ \ • " "/a < ---------------------------- ■ (5-46) sm (7 ) sin (By) The Cosine Law II for AZ B X and AZAY yields cos (Bz) = cosh (d (X, B)) sin (f3) sin (9X ) — cos (f3) cos (Bx) , (5.47) cos (Q z) = cosh (d (Y, A)) sin (a) sin (9y) — cos (a) cos (6y) . (5.48) Substituting the expressions of cosh (d (X, B)) and cosh (d (Y, A)) in Eqs. 5.45 and R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 120 5.46 into Eqs. 5.47 and 5.48 yields /a \ _ (cos(/?)cos(q)d-cos(a))(cos(7 )cos(0a ;)+cos(0j,))— (1 — c o s 27 ) c o s ( /5 ) c o s ( 0 ;e) cos (9Z) — 2 (sm (7 )) 2 \ / 2 cos a cos 0 cos 7 + cos2 a + cos2 0 + cos2 7 — 1 (sin (7 )) cos (9Z) = • a / 2 cos 7 cos 9X cos By 4 - cos2 7 + cos2 9X d - cos2 9y — 1, ( 5 .4 9 ) ( c o s ( 7 ) c o s ( q ;) + c o s ( / 5 ) ) ( c o s ( 7 ) c o s ( 0 3/) + c o s ( 0 ;z :) ) — (1 — cos27)cos(a)cos(/9j/) (sin (7 ) ) 2 ------------ 2 \ / 2 cos a cos 0 cos 7 + cos2 a + cos2 0 + cos2 7 — 1 (sin (7 )) • a/2 cos 7 cos 9X cos 9y + cos2 7 - + - cos2 9X + cos2 9y — 1. ( 5 .5 0 ) Equating the two expressions for 9Z in Eqs. 5.49 and 5 .5 0 yields (cos acos 9y— cos /3cos ^d-cos acos qcos 9X+cos j3cos qcos 9y+ 2 cos /3cos2 qcos 0X ) = (cos j3 cos 9X— cos acos ^d-cos acos qcos 9X+ cos /3cos qcos ^ d -2 cos acos2 qcos 9^ . That is, 0 = (2 cos a) (l — cos2 7 ) cos 9y — (2 cos 0) (l — cos2 7 ) cos 9X, 0 = 2 (l — cos2 7 ) (cos 0 cos 9X — cos a cos 9y) Therefore, either cos 7 = 1 or 0 = cos 0 cos 9X — cos a cos 9y, cos 9X cos By cos a cos 0 By symmetry (in case a,/3,q 0 0), Eq. 5.51 yields the following result: cos 9X cos 9y cos 9Z cos a cos 0 cos 7 (5.51) (5 .5 2 ) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 121 Eq. 5.52 might be called Cosine Rule for the period 3 orbit triangle. Next, cosh (d (Z, A)) = cosh (c — d(Z,B)) = cosh (c) cosh (d(Z, B)) — sinh (c) sinh (d (Z, B)) (5.53) The Cosine Law II for AZAY and A Z B X yields , , llr7 ... cos a cos 9Z + cos 9V /r r cosh (d(Z,A)) = -------; ---- A—: ----- A (5.54) sm a sm 9Z . , . cos (5 cos 6Z + cos 0X cosh (d{Z,B)) = -------, (5.55) sm p sm 0Z . m \ \/2 cos cos 9X cos 0* + cos2 f} + cos2 6X + cos2 9Z- 1 . . smh (d (Z, B )) = ------------------------_ _ _ _ _ ------------------------- . (5.56) sm (p) sm (0Z ) Therefore, substituting Eqs. 5.54 through 5.56 into 5.53 yields cos /3 cos 9Z + cos 9X cos a cos 9Z + cos 9y (cosh (c)) sin j3 sin 9Z sin a sin 9Z \/2 cos B cos 9X cos 9Z + cos2 B + cos2 9X + cos2 9Z — 1 = (smh (c ) ................................:..r,r ......— ........... .........• (5.57 sm (p) sm (6Z) Substituting the expression for cosh (c) and sinh(c) from the Cosine Rule II for A ABC into Eq. 5.57 yields the following expression: (cos a cos cos 7 ) (cos cos 9z+cos 6X) — ( l — cos2 0 ) (cos a cos 9Z+ cos 6y) = y / 2 cos a cos /3 cos 7 + cos2 a + cos2 fd + cos2 7 — 1 • yj2 cos fl cos 9X cos 9Z + cos2 /? + cos2 9X + cos2 9Z — 1 (5.58) From the Cosine Rule for the period 3 orbit triangle, cos 9X and cos 9y can be R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 122 expressed as cos 9Z cos tix = cos a cos 7 „ cos 9Z _ cos = cos p. v cos 7 Squaring up Eq. 5.58 and substituting the expressions for cos6 h and cos 9y yield (2 cos a cos /3 cos 7 + co s2 a + cos2 /3 + co s2 7 ) ( l — co s2 0) co s2 9Z = ( l — cos2 /?) cos2 7 . Therefore, c o s 2 Qz = . ( 5 .5 9 ) * (2 cos a cos (3 cos 7 + co s2 a + co s2 (3 + co s2 7 ) ' Clearly, 9Z, and by th e sam e ta k en c o s9X, cos 9y, are u n iq u ely defined on ce th e in cid en ce an gles are set eq u al to th e reflection an gles, in w h ich case th e y are giv en by n „ co s2 a cos 9X = -------------------- — — — ------- --------— , (5.60) 2 cos a co s p co s 7 + cos^ a + co s2 (3 + co s2 7 cos2 6y = ----------------------- C 0 S ^------- 2- 2 ■ ■ ■ (5.61) 2 cos a cos (3 co s 7 + co s2 a + co s2 p + co s2 7 Observe that the expression for 9Z is consistent with the one obtained from the orthic triangle. With 9X, 9Z uniquely defined and f3 specified, AZ B X is uniquely defined and so are its sides x, c — z. A similar argument yields uniqueness of y and Hence the inscribed triangle with its incidence angles equal to the corresponding reflection angles is unique and is hence the orthic triangle. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 123 5.6.5 Second order variation In this section, the Hessian matrix of F (x , y, z) is computed at the critical points. It will be proved that this Hessian matrix is positive definite and hence that the critical point is a local minimum point. To compute the Hessian matrix, the secon d order p artial derivatives o f d (x , y ) , d (y, z) , an d d (z, x) a t th e critical p oin t are co m p u ted as follows: d 2 d * d(x'v) d / sin h ( —a + x ) co sh y + co sh ( —a + x ) sin h y sin 7 \ d x \ sin h (d (x , y )) J (cosh ( —a + x ) co sh y + sin h ( —a + x ) sin h y sin 7 ) sin h (d (x, y ) ) (sin h ( —a + x ) co sh y + cosh ( —a + x) sin h y sin 7 ) . .. d . ■ — — - —— —— — — --------------- — o----------------------------co sh (a (x.y)) — a (x, y) (sinh (i(x,y))f '• y ,y” dx 1 1 (sin h (—o + o ;)c o s h y + c o s h ( —a + a :)s in h y c o sy )2 cosh (d(x, y ) ) tanh (d (x, y)) (sinh (d(x,y))f sinh (d(x, y)) sin2 (6X) (5.62) tanh (d [x,y)Y d2 d f cosh {—a + x) sinh y + sinh {—a + x) cosh y cos 7 dy \ sinh (d {x,y)) (cosh {—a + x) cosh y + sinh (—a + x) sinh y cos 7 ) sinh (d (x, y)) (cosh (—a + x) sinh y + sinh (—a + x) cosh y cos 7 ) , , , . . . d , . , —------ ^ -------- L ........... > 1 cosh (d (x, y)) — d (x, y) (sinh (d(x,y))) dy 1 (cosh (—a+x)sinhy + sinh (— a+a;)coshy cos7 ) 2 cosh (d (x, y)) tanh (d (x, y)) (sinh (d (x, y)))2 sinh (d (x, y)) sin2 (dy) tanh (d (x, y)) ’ ’ (5 .6 3 ) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 124 aidyd(x'v) d ( sinh (— a + x) cosh y + cosh (— a + x) sinh y cos 7 dy \ sinh (d(x,y)) (sinh (— a + x) sinh y + cosh (— a + x) cosh y cos 7) sinh (d (x, y)) (sinh (— a + x) cosh y + cosh (— a + x) sinh y cos 7 ) u A ( w ® j / \ (sinh (d (x, y) ) ) 2 _ _ ^ (d (S’V)) W (S’V) (— sinh (a— x) sinhy+cosh (a— x) coshy cos7 ) , . cosh (d(x,y)) , . sinh (d(x,y)) +cos x g cos v (— sinh (a — x) sinh y + (cosh (d (x, y)) + sinh (a — x) sinh (y) cos ( 7 ) ) cos 7 ) sinh (d (x, y)) ) sinh (a — x) sinh y + cosh (d (x, y)) cos 7 + sinh (a — x) sinh (y) cos2 ( 7 ) sinh (d (x,y)) cosh(d(x, y)) C O S 7 —sinh(a— x) sinh(y) sin2 ( 7 ) ( M ^ cosh(d (x,y)) „ ^ ^ COS ; \ ~ 7~ j ~7 \V \ 2/J + C O S ( g ,)?+(^,y ))c o smh (d (x,y)) y sinh (d (x, y)) sinh(d (x, y)) cosh(d(x, y)) , .. . .. . . sinh (a — x) sinh (y) sin2 (7 ) in i+ f e i < C0S< « * > “ S< » ■ > + sinli(d(x,y)) cosh2 (d (x, y)) sinh (d (x,y)) cosh2 (d (x, y)) sinh (d (x,y)) sin (dx) sin (dy) sinh (d (x, y)) sin (dx) sin (dy) — sinh (y) sin (7) sin (dy) sin (8X) sin (6y) — sinh (d (x, y)) sin (dx) sin (ds (5.64) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 125 d2 w d(v’z) d f sinh (— 6 + y) cosh £ + cosh (— b + y) sinh £ cos a dy \ sinh (d (y, z)) (cosh (—b + y) cosh z + sinh (—b + y) sinh z cos a) sinh (d (y, z)) (sinh (—b + y) cosh z + cosh (—b + y) sinh z cos a) , . .. d , . — — - - . , . . ... 2 cosh (a (y, zj) d (y, z) (smh (d(y,z))) dy 1 (sinh (—b+y) cosh 2 + cosh (—b+y) sinh 2 cos a)2 cosh (d (y, z)) tanh (d {y,z)) (sinh (d (y,z)))2 sinh (d(y,z)) sin2 (6y) tanh (d (y, z)) ’ d2 d f cosh (~b + y) sinh z + sinh (—b + y) cosh z cos a ( 5 .6 5 ) dz \ sinh (d (y, z)) (cosh (—b + y) cosh 2 + sinh (—b + y) sinh 2: cos a) sinh (d (y, z)) (cosh (—b + y) sinh 2 + sinh (—b + y) cosh 2 cos a) . ,, d ,. . s ^ ( d f o , z ) ) ------------------ 1 co sh ( d fa’2)) rd <»■ *) 1 (cosh (— 6 +y)sinh 2:+sinh (—b+y) cosh 2: cos a)2 cosh (d (y, z)) tanh (d (y, z)) sinh2 (d (y,z)) sinh (d (y,z)) sin2 (9Z) tanh (d (y, z)) ’ ( 5 .6 6 ) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 126 d2 W * d{v' z) d ( sinh (—b + y) cosh 2: + cosh (— 6 + y) sinh z cos a ' dz \ sinh (d (y, z)) ) (sinh ( —6 + y) sinh z + cosh (—b + y) cosh z cos a) sinh (d (y,z )) (sinh {—b + y) cosh £ + cosh ( —6 + y) sinh z cos a) , . .. d . . ^ (d h ;;)))5 ( » . *) ( — sinh (b—y) sinh 2 + cosh (b—y) cosh 2: cos a) .cosh (d(y,z)) . . sinh {d {y,z)) +cos y ^ ^ ^ cos z- (— sinh (b — y) sinh £ + (cosh {d (y, z)) + sinh (b — y) sinh (z) cos (a)) cos a) sinh (d (y, z)) ) sinh (b — y) sinh z + cosh (d (y, z)) cos a + sinh (b — y) sinh {z) cos2 (a) .. . cosh (d (y, z)) . + C O sft)S inh ( d f a j ) ) 008^ ) /n ^ cosh {d {y, z)) /n ^ sinh (d (»,*)) 008 sinh (d (y,z )) cosh(o? (y, z)) cos a —sinh(6 — y) sinh(z) sin2 (a) . . cosh(h (y, z)) . . sinh (d(y,z)) ~^C ° y sinh(h (y, z)) z cash {d („,*)) (cos {%) cos m + cos a) sinh (b - „) sinh (z) sin2 (a) sinh (d (y,z)) sinh(h (y,z)) cosh2 (d (y, z)) . sinh (d (y, z)) cosh2 (d (y,z)) sinh (d (y, z)) sin {dy) sin (6Z) sinh (d (y, z)) sin {6y) sin (6Z) — sinh (z) sin (a) sin (9Z) sin (9y) sin (9Z) — sinh (d (y, z)) sin (9y) sin (9Z) (5.67) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 127 d2 w d(z'x) d ( sinh (— c + z) cosh x + cosh (— c + z) sinh x cos /3' dz \ sinh(h (z,x)) ) (cosh (—c + z) cosh x + sinh (— c + z) sinh x cos /3) sinh (d (z , x)) sinh (—c + z) cosh x + cosh (—c+z) sinh x cos 8 , .. d ,, . ---------------------------- -— 2 — cosh (d (z, x)) — a (z, x) (sinh (d(z,x))) ’ dz 1 (sinh (—c+z) cosh a;+ cosh (—c+z) sinh x cos fd)2 cosh (d (z, x)) tanh (d (z,x)) (sinh (d(z,x)))2 sinh (d (z,x)) sin2 (8Z) tanh (d (z, x)) ’ d2 W d (Z,x) d ( sinh (—c + z) cosh x + sinh (—c + z) sinh x cos /3 (5.68) dx \ sinh (d (z, x)) sinh (—c + z) sinh x + sinh (— c + z) cosh x cos f3 sinh (d (z, x)) (sinh (—c + z) cosh x + sinh (—c + z) sinh x cos 8) , .,, .. d , — ----------------------------------- -— 5..... - ■ ■ ■ cosh (d (z. x)) — a (z, x) (sinh (d (z, x))) K y dx K ’ J 1 (sinh (—c+z) cosh £ + sinh (—c+z) sinh x cos /3 ) 2 cosh (d (z, x)) tanh (d(z,x)) (sinh (d(z,x)))2 sinh (d (z,x)) sin2 (9X) tan h (d (z,x ))’ (5.69) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 128 i k d ( z ' x) d f sinh (— c + z) cosh x + cosh (— c + z) sinh x cos ft dx \ sinh(d (z,x)) (sinh (— c + z) sinh x + cosh (— c + z) cosh x cos j3) sinh [d (z,x)) (sinh (— c + z) coshx + cosh (—c + z) sinhxcos 8) , /7/ > . d . — ------ ----------------------------- -— o ................... - cosh (a (z, x)) — a (z, x) (sinh (d(z,x)))2 K { n dx v ; (— sinh (c— z)sinhx+cosh (c— z) coshx cos ri) . cosh(<i(z,x)) .. . —-----------■ ■ - — -------- bcos (8Z) — cos (8X) sinh (a (z, x)) smh (a (z,x)) (— sinh (c — z) sinh x + (cosh (d (z, x)) + sinh (c — z) sinh (x) cos (/3)) cos (3) sinh (d (z, x)) ) — sinh (c — z) sinh x + cosh (d (z, x)) cos /3 + sinh (c — z) sinh (x) cos2 (/?) sinh (d (z, x)) cosh(d(z,x)) cos/3— sinh(c— z)sinh(x)sin2 (8) .. . cosh(d(z,x)) . — '-cos (8Z) -— 7 7 3 7 -A cos (8X) ,, , cosh(d(z,x)) . + cos (9Z) — T - 7 zz cos (0B ) sm h(a(z,x)) . cosh (d (z, x)) . + C °S(<W i nh (d (c ,x ))C °S(fc) cosh (d (z, *)) , sinh (d (z, x)) ( l cosh2 (d (z,x)) sinh (d (z, x)) cosh2 (d (z,x)) sinh (d (z, x)) sin (9Z) sin (0*) sinh (d (z, x)) sinh (d (z, x)) sinh(d (z, x)) / m \ m \ m sinh (c — z) sinh (x) sin2 (/3) (cos (6Z) cos + cos f3)------------------------------------- sin (8Z ) sin (9X) — sinh (x) sin (/?) sin (0X ) sin (0*) sin (9X) — sinh (d (z, x)) sin (02) sin (0X ) (5.70) With those second order partial derivatives, the Hessian matrix H can be easily seen to be equal to H = Axy + Ayz + Azx (5.71) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 129 where, A X y sin 2{@x) sin(8x) sm(fly) q tanh(d(x,y)} sinh (d(x,y)) sin(8x) sm($y) sin2(8y) a sinh(d(x,y)) tanh(d(x,y)) (5.72) A yz 0 0 0 0 sin2 (fly) sin(fly) siii(fla) tanh(d(y,z)) sinh (d(y,z)) n sin(fly)sin(flz ) sin2(flz) sinh(d(y,z)) tanh(d(y,z)) sin2(flx) q sin(flz) sin ^ x ) tanh(d(z,x)) sinh(d(z,a:)) (5.73) 0 sin(fla ,)sin(fl3 ;) sinh(d(z,a;)) 0 0 0 sin2(flz ) (5.74) tanh(d(z,x)) To prove that H is positive definite, let Axy, Ayz, Azx be defined as follows: A y z A — s^-zz — tsnh{d(x,y)) sin(8X) sin(fly) sinh (d(x,y)) sin2(fly) sinh(d(x,y)) tanh(d(x,y)) sin2 (fly) sin(fly) sin(flz) tanh(d(y,z)) sinh(d(y,z)) sin(fly) sm(flj.) sin2(fle) sinh (d(y,z)) tanh(d(y,z)) sin2 (Ax) sin(flx) sin(flx) tanh(d(z,a:)) sinh(d(z,a;)) sin(fls) sin(flx) sin2 (8 z) sinh(d(^,x)) tanh(d(^, x)) To prove that Axy, Ayz, Azx > 0, it is required that 9X, 6y, 9Z are not equal to zero. Indeed, 0X = 0 would imply, by uniqueness of the geodesics in hyperbolic space, that Z — B and Y = C. The latter would In turn imply that a — /3 = f and R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 130 furthermore a + (3 + 7 > ir, a violation of the hyperbolic condition. It follows that ta^h(d(xi')) > because 0X > 0, and d (x , y) < 00 by minimality. Hence the diagonal entries of Axy, Ayz, Azx are positive. Next, observe the following: det (^AX y^ — sm „) sin2 (6 y) _ ( sin (^ )sin (9y)\ tanh (d (x, y)) tanh (d (x, y)) \ sinh (d(x,y)) J sinh (d (x, y)) = (sin (9X) sin (9y ) ) 2 > 0 Thus Axy, Ayz, Azx are positive definite. Hence H = Axy + Ayz + Azx > 0. To show that the latter is positive definite, let W \ w2 Ws w = be a vector such that 0 = wtH w, (AX y T Ayz ~r Azx) w , = wTAxyw + wTAyzw + wTAzxw. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 131 The above implies that 0 = 0 = Wi w2 Since Axy, Ayz, Azx > 0, it follows that *xy W2 W3 A ■yz W i W3 A, Therefore, the Hessian matrix w = 0. Wi w2 w2 W3 Wi w3 H = sin2 ) sin3(0x) tanh(d(x,y)) ta nh(d(z,x)) sin(03;) sin {By) sin 2(8y) sinh (d(x,y)) sin(9z)sm(9x) sinh(d(z,x)) sin(03;) sin(0w) smh(d(x,y)) + sin2 (By) tanh(d(x,y)) tanh(d(y,z)) sin(0;,) sin(0z ) sinh (d(y,z)) sin(0z) sin(gx) sinh(d(z,x)) sin(0H ) sin(0a) sinh(d(j/,z)) sin 2(9z) sm2(dz) tanh(d(j/,z)) tanh(d(z,x)) is positive definite and the second order variation test passes. 5.6.6 Fatness formula In this section, the fatness formula is derived for a geodesic triangle with acute angles only. Given that A X Y Z is the inscribed triangle that satisfies the first order variation, then the internal angles of A X Y Z at X, Y, Z are tt — 29x, n — 29y,ir — 29Z, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 132 respectively, where 9X, 8y, and 9Z are determined from Eqs. 5.59 through 5.61. cos (jt — 29 x) = 1 — 2 cos2 0X 2 cos a cos j3 cos 7 + cos2 /3 + cos2 7 — cos2 a (2 cos a cos ft cos 7 + cos2 a + cos2 j3 + cos2 7 ) ’ cos (7 r — 2 6y) = 1 — 2 cos2 By 2 cos a cos (3 cos 7 + cos2 a + cos2 7 — cos2 /3 (2 cos a cos 0 cos 7 + cos2 a + cos2 j3 + cos2 7 ) ’ cos (7r — 29z) = 1 — 2 cos2 9Z _ 2 cos a cos 0 cos 7 + cos2 a + cos2 0 — cos2 7 (2 cos a cos ft cos 7 + cos2 a + cos2 0 + cos2 7 ) The Cosine Rule II for A X Y Z yields the following results: cos (7r — 26X) cos (w — 29y) + cos (n — 29z) cosh d(X,Y) = sin (7r — 28x) sin (n — 29y) (1 — 2 cos2 9X) (1 — 2 cos2 dy) + 1 — 2 cos2 8Z (2 sin cos 0X ) (2 sin cos 0y) (2 cos2 0X cos2 9y—cos2 9X— cos2 9y— cos2 9 z+1) 2 (sin 9X cos 9X) (sin 9y cos 9y) ’ sinh d (X, Y) = yjomh2d(X, Y) - 1 y^(cos2 9X+cos2 6^ + cos2 0* — I) 2 — (2 cos 9X cos 8y cos 8^ 2 (sin 6X cos 9X ) (sin 8y cos 9y) . I/T, (2cos2 9Vcos2 9Z—cos2 9X—cos2 9V— cos2 8Z+ 1) cosh a (r, Z) — --------- —n — ~ 7T\----------- 1 2 (sm 8y cos By) (sm 9Z cos Bz) • \ /(cos2 02,+ cos2 + cos2 ^ — 1)2 — (2 cos cos 6L cos 032 sinhd (Y, Z) — - -------------- —— — ------- . , , ■ ■ ---- — r---------------- . 2 (sm By cos 9V) (sm 8Z cos Bz) , ,. _ ^ (2 cos2 9Z cos2 0^ - cos2 0X - cos2 9y - cos2 0*+1) cosh a (Z.X) = ----------- : , 2 (sm 6Z cos 9Z) (sm Bx cos Bx) \ / (cos2 8X + cos2 9V+cos2 9 z — 1)2 — (2 cos Bx cos 9V cos 62“ sin h d (y , X ) = ---------------- — 7-:— _........- "V'y: ...... r........... — — 2 (sm 8z cos 9Z) (sm 9X cos 8x) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Then ( 5 .7 5 ) ( 5 .7 6 ) ( 5 .7 7 ) ( 5 .7 8 ) , ( 5 .7 9 ) ( 5 .8 0 ) , ( 5 .8 1 ) ( 5 .8 2 ) i . ( 5 .8 3 ) 133 Therefore, sinh {d (X, Y )+ d (Y, Z) + d (Z, X)) = cosh d (X , Y) cosh d (Y, Z) sinh d (Z, X )+cosh d (X, Y) cosh d (Z, X) sinh d (Y, Z) +cosh d (Y, Z) cosh d {Z, X) sinh d (X, Y) +sinh d (X, Y) sinh d (Y, Z) sinh d (Z, X) (1— cos2 0X— cos2 9V—cos2 03 [, 77 „s2 7 7 = 777---- 2'a\ " ( ..... 2"a \ (.....2 a"\V ( cos + cos dy + cos ^ _ 1 ) ~ ( 2 cos cos°y co s#*) 2 (cos2 9X) (cos2 dy) (cos2 0Z) v Finally, substituting the expressions for cos9x,cos0y,cos0z yields the expression for the fatness as follows: sinh (d (X, Y) + d (Y, Z) + d (Z, X)) = 2\/pcosQ:cos/?cos7 +cos2Q:+cos2/3+cos27) (2 cosacos/3cos7 +cos2a+cos2/3+cos27 — 1 ) and the expression for the fatness of a geodesic triangle in a constant negative curvature k is given by the following formula: 1 Sp (AABC) = — 7= sinh \/-k ,-i ^ 2\/(2cosQ:cos/3cos7 +cos2a+cos2/3+cos27) (2 c o s q c o s /3 c o s t4 - c o s 2qM -cos2/3 + c o s27 —1)^ . Clearly, the maximum fatness triangle is obtained when a — j3 — 7 = 0, that is, the ideal triangle with its vertices at infinity. In the Poincare unit disk model, this triangle has its vertices on the unit circle, 120 degrees apart. It follows that the fatness of the Riemannian manifold with constant negative curvature k is as follows: 5p — 1— sinh-1 (aV s ) ~ — — = 2.887. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 134 1 C Figure 5.3: The Fejer construction for the obtuse angle case. The construction remains valid in hyperbolic geometry. 5.6.7 O btuse angle case In this section, the fatness for a geodesic triangle with an obtuse angle is derived via the Fejer construction. It turns out that the fatness for a geodesic triangle with an obtuse angle is equal to twice the length of the altitude corresponding to the obtuse angle. The Fejer constructions in the Euclidean and hyperbolic spaces are the same and can be given as follows: Given that A ABC is a triangle with the internal angle a > | in a Riemannian manifold with constant curvature k < 0 and A X Y Z is an inscribed triangle of A ABC where X E [BC] is fixed and Y E [CA\, Z E [AB] are such that the perimeter is minimized, then reflecting X across [CA], [AB] to get X, and X, respectively, as shown in Figure 5.3 yields the following result: d(X, Y) + d(Y, Z) + d(Z, X) = d(X, Y) + d{Y, Z) + d(Z, X ). (5.84) Recall that in a nonpositive curvature space the distance function from the point R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 135 X to an arbitrary point on [CA] is convex (see [Jos97, Chap. 2, Sec. 2.1]). Since A and Y are on the same side of the foot of the perpendicular from X to [CA], it follows that the distance function is monotone; hence d(X, A) < d(X,Y). A similar argument for AX AY yields d(A, X) < d(Y,X). Hence d(X,A) + d(A,X) < d(X,Y) + d(Y,X) < d(X,Y) + d(Y,Z) + d(Z,X) by triangle inequality. It follows that optimality is reached for Z = Y — A and Y e l c t m A B ] d { X 'Y ) + d(Y' z ) + d(Z'x ) = 2 d ( A x ) ■ (6 '88) Since d(A, X) is minimum for [AX] A [BC], then the fatness of a geodesic triangle with an obtuse angle is twice the length of the altitude corresponding to the obtuse angle. Given that k < 0, then from the Pythagoras’ theorem for AABX, and the Cosine Rule II for A ABC, it follows that sinh (d (A, X)) = sin (3 sinh c — sin /3 V cosh2 c — 1 = v (2 cos a cos fi cos 7 + cos2 a + cos2 j3 + cos2 7 — 1). sm a Therefore, the fatness formula for a geodesic triangle with obtuse angle at a is given by 5f (AABC)— -sinh-1 (— - —\ / (2 cos a cos (3 cos 7 + cos2 a + cos2 (3+cos2 7 — 1)) . s j tv ySlH O t J R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 136 Since the billiard dynamics is not defined when the ball hits an angle at the boundary of the table this obtuse case does not have such an interpretation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 137 Chapter 6 Random graph and hyperbolic graph The theme of this chapter is to determine whether the internet and WWW are hyperbolic graphs. Due to the size and complexity of the internet and WWW, it is almost impossible to compute directly the 8 of the internet and WWW. Since the internet and WWW have traditionally been modeled as random graphs, this problem can be partially solved by computing the 8 for random graph models. In this chapter, the connections between random graphs and hyperbolic metric spaces are discussed. This chapter involves the following graph models: First, the random graph as modeled by Erdds and Rbnyi; second, the small world graphs as modeled by Watts and Strogatz; third, the scale free networks as modeled by Barabasi- Albert. Finally, the relation between random graphs and hyperbolic metric spaces is developed. 6.1 Random graphs m o d e le d by Erdos and Renyi The material in this section follows [BolGl] and [Pal85] closely. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 138 The theory of random graphs was developed by Paul Erdos and Alfred Rhnyi. In the Erdbs-Renyi approach, probabilistic methods were used as the underlying treatments of random graphs. Intuitively, a random graph is such that the number of vertices, edges, and connections among them are specified in a random way. For each natural number n, the set of all graphs G of order n such that Vq = {1,2,..., n} is denoted by G (n). The two simplest models of random graphs of order n are as follows: 1 . G (n, m) model: Given positive integers n and m where 0 < m < (”), then the G (n, m) model is contained in G (n) and consists of all labelled graphs of order n and of size m, where each graph has the same probability. Therefore, G (n, m) has elements and the probability of each graph G £ G (n, m ) is given by graphs of order n and is such that each edge is chosen independently with probability p. Therefore, the probability of G £ G (n , p) with m edges is given D efinition 72 Given that iln is a model of random graphs of order n, then almost 2. G(n,p) model: Given a positive integer n and a real number p where 0 < p < 1, then the G (n,p) model is contained in G (n) and consists of all labelled by p i g ) = pm{i -pfiy™. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 139 every random graph in Qn has a property Q in the sense of Erdds-Renyi, if the probability of graphs with this property approaches 1 as n — > oo. D efinition 73 A property Q is monotone increasing if for every G £ Q and G C H, then H £ Q. The notation G C H means that G £ G (n) is a subgraph of H £ G(n) . Observe that G and H have the same order. Definition 74 The property Q is convex if for every F C G C H and F £ Q ,H £ Q, then G £ Q. The relationship between the G (n, m) and the G (n,p) models is stated in the following theorem. Theorem 48 Given that m — m(n) is any sequence of positive integers such that < m < ( l + e ) p Q where e > 0 is fixed, then if almost every graph has a property Q in the G (n , m ) model, then so is it in the G (n, p) model. Given that the property Q is convex and m (n) = [pQ)J , then if almost every graph has a property Q in the G (n,p ) model, then so is it in the G (n , m ) model. Proof. See [Pal85, Appendix 6 , Thm 6.1]. ■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 140 T heorem 49 In the G(n,p) model, given that p is constant, then for every fixed integer k > 0, almost every graphs has the following property: 1. Almost every graph has diameter 2. 2. Almost every graph is k-connected. 3. Almost every graph contains a given subgraph of order k as an induced graph. 4. Almost every graph is nonplanar. Proof. It follows from the applications of [Pal85, Chap. 2, Thm. 2.3.1]. ■ From this previous theorem, the G(n,p) model with fixed p and the G (n, m) model with m(n) — [pGDJ can be considered as hyperbolic spaces. In general, random-graph theory studies the properties of random graphs of order n as n — > oo where m = m(n) and p — p{n). A construction of a random graph can be considered as an evolutionary pro cess. Given a graph with n isolated vertices, the graph developed by successively adding random edges as p or m increases eventually becomes a fully connected graph. Random graph theory studies the relationship between the property Q and the parameter p or m of random graphs such that the property Q arises. Erdos and Renyi discovered that many monotone-increasing properties of random graphs appear suddenly i.e., graphs of a size slightly less than a certain threshold are very unlikely to have property Q, whereas graphs of a size slightly larger than the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 141 threshold are almost certain to have this property. A formal definition of threshold function is given as follows: D efinition 75 Given that Q ,n is a model of random graphs of order n and Q is a property of graphs, then t = t(c, n) is a threshold function for property Q if there is a number cq such that for G e f i n with p(n) — t (c, n) in G (n, p) or m(n) ~ t (c, n ) in G (n , m) 1. if c> C q , almost all graphs G have the property Q. 2. if c < C o , almost all graphs G do not have the property Q. The following theorem shows that the threshold function for connectivity is the function p (n) = p p in G (n, p) or m(n) ~ |n log n in G (n, m ). T heorem 50 Given that G 6 G (n,p ) withp = c p p orG £ G (n , m) with m (n ) ~ c\n log n, then 1. if 0 < c < 1, almost every graph G is disconnected. 2. if c > 1 , almost every graph G is connected. Proof. See [Pal85, Chap. 3, Thm. 3.1.1]. ■ D efinition 76 Given that Q is a monotone increasing property, then the function pc (n) is a threshold function for Q in G (n,p(n)) if 0, 1, P (jl) I then Pn,p(n) i.Q ') n— * o o pc (n) , 1 , then P„,p(n ) (Q) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. lim n— > 0 0 m c (^7lj 142 where Pn,p(n ) (Q) denotes the probability that a graph in G (n,p(n )) has property Q. Similarly, the function m c (n) is a threshold function for Q in G (n , m (n)) if m (7 1 ) j then Pn,m(n) i.Q ') '' 0, I O O , then Pn,m(n) (Q) * where Pn,m {n) (Q) denotes the probability that a graph in G (n, m (n)) has property Q. The following theorem defines the threshold function for the diameter of a graph. T heorem 51 Given thatp (n )2 n —21ogn — > 00 and n2 (1 — p (n)) — >00 inG (n,p) or 0 < m (n ) < (”) and 2 ^ ^ lo g n — ► 00 in G (n , m ) , then almost every graph has diameter 2. Proof. See [BolO l, C hap. 10, C or. 10.11]. ■ Theorem 52 Given a function d (n ) > 3, — 3 lo g lo g n — ► 00, then if 1. 0 < p (n ) < 1 satisfies p (n )d^ nd^ ~ l — 2 lo g n — > 00 p (n )d^ -1 n d(nP 2 — 2 lo g n — > —00 /o r the G (n ,p ) model or 2. 0 < m(n) < (2) satisfies 2 < i(n )-im (77,)^") n ” d(n)_1 — lo g n — > 00 2d(n)"2m (n )d^ _1 n ~ d^ — log n — » —00 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 143 for the G (n , m) model, then almost every graph has diameter d. Proof. See [BolOl, Chap. 10, Cor. 10.12]. ■ To find the degree distribution of the graph, the number of vertices of degree k is considered. The following theorem shows that the number of vertices of degree k follows a Poisson distribution. T heorem 53 Given that e > 0 is fixed, k is a fixed nonnegative integer, X k is a 3 3 number of vertices of degree k, en~2 < p (n) < 1— en~z, and Xk (n) = n ^ n ^ (n f (1 - p (n))ri“1 ''f c , then the following assentations hold: 1. If lim^oo A f c (n) — 0, then (X k = 0) = 1. 2. J/lim ^oo A k (n) = 0 0 , then lirrin-^^P (X k > t) = 1 for every fixed t > 0. 3. If 0 < liminfn_ + 0 oAk (n) < lim supn_> c o (n) < 0 0 , then X k converges in distribution to a Poisson random, variable with parameter Xk, that is P (Xk = r) -* exp (— A f c ) for every fixed r. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 144 Proof. See [BolOl, Chap. 3, Thm. 3.1]. ■ Since the Poisson distribution decays rapidly for large value of k, Xk does not deviate much from its average E (Xk). It follows from [AB02, Eq. 14] that if the vertices are independent, then the degree distribution of a random graph can be approximated as follows: P(k) = — w (6.1) n n = ( ” ^ (1 ~ p (n))n_1_fc • For the large value n, and np (n) — ^-constant, the degree distribution can be ap proximated by a Poisson distribution P (k) « exp (-np) (6.2) 6.2 Small worlds graph modeled by Watts-Strogatz The material in this section follows [Wat99] closely. The "Small World" phenomenon is the idea that every two individuals in the world can be reached through a short chain of social acquaintances. Small-world phenomena have been observed for several networks especially in the social science. The six degrees separation indicates that a class of small-world graph is highly clustered with small characteristic path lengths. Watts and Strogatz [WS98] noticed this feature on numerous networks in Nature. The random graphs as modeled by Erd6 s-Renyi do not provided the high cluster with small path length. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 145 Watts and Strogatz [Wat99, Chap. 3, Sec. 3.1.1 and Sec. 3.1.2] proposed the a- and /3-models that generate graphs with the small world property. Intuitively the a-model is motivated by how people actually make new acquaintances in a real social network. In contrast, the /3-model is motivated by simplicity of construction and removes the reference to social network. 6.2.1 a-m odel The concept of a-model is to construct a graph which captures the natural connec tions in a social network. The a-model is designed to represent a graph that lies between two extreme models: the caveman and Solaria world. The caveman world represents the model with the property that everybody you know knows everybody else you know and no one else. In contrast, the Solaria world represents the model with the property that the influence of current friendships over new friendships is so slight as to be indistinguishable from random chance. Given that n is the number of vertices, k is the average degree, and a G [0, oo] is a tunable parameter, then the a-model, which is a graph of order n and of size , can be constructed as follows: 1. Randomly select vertex i from n possible vertices, and for every j ^ i, compute a measure Rij of vertex Vs propensity to connected to vertex j. Rij is equal to 0 if vertices i and j are already connected, otherwise Rij can be computed R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 146 by the following formula: Rij = P, m j = 0 (1 —p)+p, 0 < rriij < k (6-3) 1 , rriij > k where rriij is the number of vertices which are adjacent to both i and j and p < C Q) 1 is a baseline random probability of an edge (i,j) existing. 2. The probability that vertex i will connect to the vertex j is given by P * = (6-4) j^i Then randomly select vertex j ^ i according to the probability Pij and con nect vertex i to j. 3. Repeat step 1 with the restriction that if the vertex i is chosen, it cannot be chosen again until all other vertices have been chosen. This procedure is repeated until the graph consists of edges. 6.2.2 /3~model The /3-model is a one-parameter model that lies between an ordered finite dimen sional lattice and a random graph. In fact, the /3-model has properties similar to the a-model without the concept of social network. When /3 varies from 0 to 1, the corresponding graph changes from a regular lattice graph to an approximately R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 147 random graph in the G (n, m) model with m = |_fc|J. Given that n is the number of vertices, k is the average degree, and /3 £ [0, 1] is a tunable parameter, then the /3-model, which is a graph of order n and of size J , can be constructed as follows: 1. Start with a ring lattice graph of order n in which every vertex is connected to its first k nearest neighbors ( | on either side). In order to have a connected graph at all times, the graph should have n 3> k Inn 1. 2. Each vertex i is chosen in turn, along with the edge that connects it to its nearest neighbor in a clockwise sense (i, i + 1). Each (i,i + 1) edge is ran domly rewired according to the probability j3. If the (i,i + 1) edge is selected to be rewired, then the (i, % + 1) edge is deleted and rewired such that i is connected to another vertex j, which is chosen uniformly at random from the entire graph (excluding self-connections and repeated connections). 3. Repeat step 2 until all vertices have been considered once, then the procedure continues for next nearest neighbor (i, i- 1-2) edges, and so on. The procedure is completed until all edges in the graph have been considered for rewiring exactly once. In can be seen from [BW00] that the shape of the degree distribution of the Watts-Strogatz /5-model is similar to that of the random graph which has a peak at k and decays exponentially for large k. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 148 6.3 B arabasi-A lbert scale free network The material in this section follows [AB02] and [DM02] closely. Recently, there has been a lot of study suggesting that complex real-world net works (for example an internet) are following neither the random graph models nor the Watts-Strogatz models in the sense that the degree distribution of real world networks significantly deviates from the Poisson distribution [FFF99]. In fact, most complex real-world networks have their degree distributions follow the power-law tail i.e., p(k) ~ Ar7, where 7 is independent of the scale of the network. Such a network is called scale free. Barabasi and Albert ([BA99] and [BAJ99]) argued that the scale free networks can be generated by a combination of two mechanisms: growth and preferential attachment. Most of the complex real-world networks can be described as open systems where new vertices can be added to the networks throughout the life of the networks. In addition, the new vertices are likely to be connected to vertices of higher degrees. The algorithm for the Barabasi-Albert model is given as follows: 1. Growth: Given a graph Go of small order no and of size mo, then Gt+1 can be recursively obtained from Gt by adding a new vertex with I (< no) edges to Gt such that the new vertex is connected to I different vertices in Gt. 2. Preferential attachment: The I different vertices in Gt can be chosen such that the probability II (*) that a vertex i is connected to a new vertex depends on R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 149 the degree h of the vertex i, i.e., (6.5) o After t time steps, the resulting graph Gt is a graph with n = t -F % vertices and It + mo edges. In addition, the degree distribution P (k) can be computed using continuum, master-equation or rate equation approach. All of these approaches provide the same asymptotic degree distribution P (k) ~ fc -7 ,7 = 3 and is inde pendent of I. In the continuum approach proposed by Barabasi-Albert [BAJ99], the degree distribution P (k) can be computed by calculating the time dependence of the degree for a given vertex i with the assumption that at t time steps the probability that the vertex i is added at time Tj is given by Then the degree distribution P (k) at time t can be computed by the following formula: where, in the limit as t — > 00, (6.6) P (k) = 2l2k~3. (6.8) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 150 Therefore, in the continuum approach, the degree distribution follows the power law tail with the exponent 7 — 3. In the master-equation approach proposed by Dorogovtsev et al. [DMS00], the degree distribution P (k ) can be computed by calculating the difference equation of the probability that at time t the vertex i introduced at time t, has a degree k. Then the degree distribution P (k) can be computed by the following formula: In the rate equation approach proposed by Krapivsky et al. [KRL00], the degree distribution P (k) can be computed by calculating the average number of vertices with k edges at time t. The asymptotic limit of the degree distribution P (k) yields the same formula as in the master-equation approach. Given that Qt is a probability space of random graphs generated from the Barabdsi-Albert model, then the diameter of the Barabasi-Albert models can be shown, asymptotically to be log n, for 1 = 1 and for I > 2. Precisely, T heorem 54 Given that £ > 0 is fixed, then the following assentations hold. 1. I fl = 1, then a.e. Gt € Qt is connected and has diameter satisfying (a -1 — e) logn < diam (Gt) < (a -1 + e) logn where a is a solution of a exp (1 + a) — 1. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 151 2. If I > 2, then a.e. Gt G Gt is connected and has diameter satisfying (1 - e) < diam (Gt) < (1 + e) log log n log log n Proof. See [BolOl, Chap. 10, Tkm 10.29] and [BR]. ■ 6.4 Simulation The purpose of this simulation study is to compare the hyperbolic property of sev eral random graph generators defined in the previous section. Each random graph generator generates a finite graph where each edge has unit distance without direc tion. The 8q of the Gromov product, which is computed from the 4-point condition, can serve as the measure of hyperbolicity. However, the numerical simulation can only create finite graphs all of which have finite diameter and hence have finite So- Therefore, every graph can be considered as a hyperbolic metric space. Observe that the hyperbolic property would manifest itself only if the 8G is significantly smaller than the diameter of graph. It follows that the mathematical expectation of the normalized delta E (^ f^ ) , where diam is the diameter of the graph, is the key hyperbolic measure for random graph generator. The simulation methodology is set up as follows: The random graphs as modeled by Erd6s and Renyi considered here are the G (n, m ) models. Observe that the connectivity of G (n, m) depends on the number of edges; hence to avoid disconnected graphs in simulation, the G (n, m) model is R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 152 generated based on the following procedure: First, the backbone graph which is the connected graph of order n and of size mo is generated; then the rest of the m — mo edges are randomly selected from the Q) — mo available edges. In fact, the backbone graph can be a deterministic or a random graph of order n. Given that the vertex set of the backbone graph is {1,2, ,n} , then the 4 backbone graphs in the simulation are defined as follows (see Figure 6.1): 1. Line backbone: Gune (n) is a graph in which E GLine(n) = {(i,i + l) :i = 1,. 2. Ring backbone: Gmng (n) is a graph in which E Gm ng(n) = {(b i + 1) mod n : i = l,...,n}-, 3. Star backbone: Gstar (n) is a graph in which E Gstar(n) = {(I5 i) ■ i — 2,..., n} , 4. Random tree backbone: GRam 2 (n) is a graph Gn obtained from the evolution where G\ is a single vertex 1 and Gt is recursively obtained from Gt-1 by adding a new vertex t to Gt~ 1 and a new edge from a vertex randomly selected from {1,... ,t — 1} to vertex t. Clearly, Gunn («), Gstar (n), and GRand (n) are trees of size n — 1 and (n) is of size n. Hence the G (n, m) models with line, star or random tree backbone have R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 153 i s L i n e b a c k b o n e R in g b a c k b o n e S t a r backbone Random tree backbone Figure 6.1: The backbone graphs in the simulation. m — n + 1 edges randomly selected from (£) — n + 1 possible edges and G (n, m) models with ring backbone have m —n edges randomly selected from (”) —n possible edges. The small worlds graphs as modeled by Watts-Strogatz considered in the simu lation are the /3-models where the j3 parameter varies as 0,0.1,..., 1. Observe that as (3 varies from 0 to 1, the random graph varies from regular graph to approxi mately purely random graph. The resulting graphs are of order n and of size . In the simulation, the effects of the parameter (3 on the hyperbolicity as well as the average degree k are observed. The scale free graphs in the simulation are generated by the Barabdsi-Albert approach in which the starting graphs are line, ring, star, and random tree back bones with no vertices. The graphs are continuously evoluting from the previous graphs by the growth and preferential attachment until the resulting graphs are of order n. To study the effect of preferential attachment on hyperbolicity, graphs generated from growth and uniform attachment are also considered in the test bed. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 154 In contrast to the preferential attachment, in the uniform attachment, the proba bility II (i) that a vertex i is connected to a new vertex is equal among all vertices 1,.,., i — 1. The degree distribution P (k) at time t for a random graph with growth and uniform attachment can be computed by the following formula: p{k)='Uiwe x p {1 ~i)' lw o ) This formula has been derived from the continuum approach by Barabdsi-Albert [BAJ99]. As t — * oo, P{k) = j exp ^1 - , (6.11) hence the uniform attachment provides an exponentially decaying degree distribu tion which depends on the parameter I of the graphs. In the scale free and growth with uniform attachment random graph generators, the resulting graphs are of order n and of size no — 1 + (n — nQ ) m for line, star, and random tree backbones and no + (n — no) m for ring backbone. In each random graph generator, each model generates a different topological graph structure and has different parameters. To understand the effect of random graph generator on hyperbolicity, the parameters for each generator are determined so that the resulting graph for each generator is of the same order and approximately the same size. In the simulation, the average degree k in small world generator is approximately twice the number of edges m for each additional vertex. Then the random graphs from different generators all have approximately the same size. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 155 Hence in this simulation the parameter k is set as k = 2,..., 2uq in the small world generator, and the parameter m is set as m = 1,..., no in the scale free and growth with uniform attachment random graph generators. The total number of edges in the Erdds and Rbnyi random graph generators is set to be the same as the size of the random graphs generated by the other generators so that the comparison among all random graph generators can be made. In this simulation, the total number of vertices n is set to 50 and 100 and the parameter no are equal to In addition, the number of simulations is equal to 100. The simulations reveal the following conclusions: 6.4.1 Random graph as modeled by Erdos and Renyi Although the random graph generators in this simulation construct random graphs on top of the backbone graphs whereas the random graphs generated by Erdds and Renyi are purely random graphs, there are no significant deviation in the degree distribution and density functions between these two methods, except in the star backbone as shown in Figure 6.2 through Figure 6.3 where the parameters for these generators are n = 3000, and m — 147,550 for random graphs with Ring backbone, m = 147,549 for random graphs without backbone and random graphs with Line, Star, and Random tree backbones. The deviation of the star backbone from the other cases follows from the fact that the star backbone has a center vertex that connects to all other vertices. This provides the existence of a vertex with high R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 156 - - - j ........................ - - - I I i ! I , i f' 1 1 1 / i ; ; : . ........................ ... ■ ■ ■ ' - ~ - - i ■ ■ - .................. . . i ..................... . ....................... y - . .. .- ~ 1 ; .’ \ ! 1 1 ' 1 1 — Random graph Line backbone Ring backbone Random tree backbone _ ...................1 . .. 1 / L 1 1 : . : : 1 !,■ i i 60 70 80 90 T O O 110 120 130 140 degree Figure 6.2: The degree probability distribution function that results from the numerical simulation of random graphs with different backbones. ’ rr S 0.6 1 i I ° - 5 ^ 0.2 i - ; 0.1 j i o it. Figure 6.3: The degree probability distribution function that results from the numerical simulation of random graphs with Star backbone. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 157 0.05,................. ,---------- 0.045;- 0.04 i- 0.035 0.03 ;r r ■ Random graph e Line backbone * Ring backbone Random tree *hl . 130 140 Figure 6.4: The degree probability density function that results from the numerical sim ulation of random graphs with different backbones. ; ; : ; e ----- -j-------+------------- ......- 1 - - .... 'T '... « : : ' c ; ; ; l 1 ■ \ . ! i : I : : ; i ' Q i J L * i | | . ■ ■ „ . . . r-.. ■ -r -T -..-.. 0 500 1000 1500 2000 2500 3000 3500 4000 degree Figure 6.5: The degree probability density function that results from the numerical sim ulation of random graphs with Star backbone. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 158 Line backbone Ring backbone Star backbone Random tree backbone 50 100 150 200 250 300 350 400 450 500 number of edges Figure 6.6: Comparison of E (5g) for random graphs of order 50. degree which contradicts the Poisson degree distribution. In the other cases, the degree distribution for the random graphs generated by this method are roughly the same as in the Erd6s and Renyi model which theoretically has an exponential decay. The E (Sq) and E (^f^) for random graphs with several backbones are shown in Figures 6.6, 6.7 for n = 50 and in Figures 6.8, 6.9 for n — 100. In the random graphs generated with different backbones, as m increases, the ex pected delta and the expected normalized delta are approximately the same among these backbones, except for the star backbone. As the number of edges increases, the expected delta decreases to 1. However the diameter of the random graphs de creases. This yields an increase of the expected normalized delta. After increasing R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 159 | 0 - 3 |- | 0.25 S I 0.2 U J 0.15 0.1 - • Line backbone i Ring backbone j Star backbone • Random tree backbone i 50 100 150 200 250 300 350 400 450 500 number of edges Figure 6.7: Comparison of E for random graphs of order 50. 151 • 1 0 1 - Line backbone Ring backbone I Star backbone j Random tree backbone i 0 200 400 600 800 1000 1200 1400 1600 1800 2000 number of edges Figure 6.8: Comparison of E (5 a) for random graphs of order 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 160 0.4 - 0.35 I 0.2 h Line backbone ' Ring backbone ! Star backbone j Random tree backbone I 800 1000 1200 1400 1600 1800 2000 number of edges Figure 6.9: Comparison of E for random graphs of order 100. the number of edges from the Line, Ring, and Random tree backbone graphs, the minimum expected normalized delta occurs at a number of edges roughly equal to 7 times the number of vertices. This suggests a network parameter that yields a good hyperbolic graph. 6.4.2 Small worlds as modeled by W atts-Strogatz The degree distribution and density functions for the small world graphs with the ft parameter varying from 0 to 1 are shown in Figures 6.10 and 6.11, respectively where the parameters for these generators are n — 3000 and m = 150,000. As the parameter ft varies from 0 to 1, the larger the tail of the density functions. The E {8q) and E for the small world graphs with the ft parameter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 161 § 0.5 / » « 5 H t < 80 85 90 beta=0.1 I beta=0.2 beta=0.3 beta=0.4 beta=0.5 beta=0.6 beta=0.7 beta=0.8 beta=0.9 beta=1.G i 100 105 110 115 120 125 Figure 6.10: The degree probability distribution function that results from the numerical simulation of small world graphs with different parameter /3. g 0.06- ■ beta=0.1 j beta=0.2 ! beta=0.3 ! beta=0.4 i beta=0.5 ( beta=0.6 beta=0.7 beta=0.8 beta=0.9 bata=1.0 r l i j v ' ’ m 8 5 100 105 110 115 Figure 6.11: The degree probability density function that results from the numerical simulation of small world graphs with different param eter /?. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 162 12j>- : " O " 3 6l - I i i 2 i 4 - ? 4 i 0 - 50 number of edges Figure 6.12: Comparison of E (8q) for small world graphs of order 50. varying from 0 to 1 are shown in Figures 6.12, 6.13 for n = 50 and in Figures 6.14, 6.15 for n = 100. The simulation shows that as (3 is increasing from 0 to 1, the expected delta is decreasing. However, the diameter of a graph is not monotonically varying with the parameter /3. Given that (3 > 0 and fixed, then after increasing the number of edges, the expected normalized delta reaches a minimum of about 0.33. The number of edges at the minimum expected normalized delta depends upon the parameter (3. The larger the parameter {3, the smaller the number of edges at the minimum expected normalized delta. After continuously increasing the number of edges, the delta of a graph will reach its minimum. Finally, after continuously increasing the size of a graph, the diameter of a graph is decreasing. This result yields an ; j ■ ® beta-0.0 | j i i * - bata=0.1 i . i ! * beta=0.2 ! 1 i ♦ beta=0.3 : 1 ' beta=0.4 1 ! beta=0.5 • 1 ■ beta=0.6 1 . bata=0.7 1 ! beta-0.8 ■ • beta=0.9 'j i beta=1.0 j R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 163 0.5 ------ 0.45 - 0.4'- - ■ i 0.35 I 0 . 2 r -j- o.osj- beta=0.0 ; beta=0.1 | beta=0.2 i beta=0.3 I beta=0.4 - beta=0.5 ] beta=0.6 ; beta=0.7 , beta=0.8 beta=0.9 beta~1.0 250 300 number of edges Figure 6.13: Comparison of E for random graphs of order 50. «- beta=0.0 i beta=0.1 J » beta=0.2 i » beta=0.3 beta=0.4 beia=0.5 beta=0.6 | 1 ; - v beta=0.7 ! 1 j beta=0.8 ] 1 ] ■ beta=0.9 j ' S _ ! • g ' 600~ ~ 800 1000 1200 number of edges 1400 1600 1800 2000 Figure 6.14: Comparison of E (Sq) for small world graphs of order 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 164 0-451- ................. - .. .f ;" 4 .-T. . I 0 ,2 U J 0.15- » • • • * ■ • ■ ' s - - - b- beta=0.0 « beta=0.1 * beta*0.2 • * beta=0.3 beta=0.4 beta=0.5 — beta-0.6 -• • beta=0.7 beta=0.8 beta=0.9 beta=1.0 1800 2000 number of edges Figure 6.15: Comparison of E { & ) for random graphs of order 100. increasing expected normalized delta. This is a behavior similar to that of the previous random graph generator. 6.4.3 Barabdsi-Albert scale free network The degree distribution and density functions for the Barabasi-Albert scale free graphs with different backbones are shown in Figures 6.16 and 6.17, respectively, where the parameters for these generators are n — 3000, and m = 147,550 for random graphs with Ring backbone, m = 147,549 for random graphs with Line, Star, and Random tree backbones. The E (Sq) and E for the scale free graphs with different backbones are shown in Figures 6.18, 6.19 for n = 50 and in Figures 6.20, 6.21 for n — 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 165 ! — Line backbone ; Ring backbtxw ! Star backbone ' .. /1 i Random tree backbone 1 ; - L .4 -............ i ; • I i 1 : i i ; i .. ! .......................... : ' 1 ............... „ - 1 ....................1 . - 1 ...................... „ - - 1 1 1 1 1 V ' " 100 200............... ’ 300 ' 400 500” ‘ ” 500 700 Figure 6.16: The degree probability distribution function that results from the numerical simulation of scale free graphs with different backbones. 0.03 j- | 0.025j 1 0.021 ■ 8 ! 0.015 r f Line backbone • Ring backbone . : Star backbone | Random tree backbone 100 200 300 400 degree Figure 6.17: The degree probability density function that results from the numerical simulation of scale free graphs with different backbones. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 166 Line backbone j Ring backbone i Star backbone i Random tree backbone j 50 100 150 200 250 300 350 400 450 number of edges Figure 6.18: Comparison of E (Sq) for scale free graphs of order 50. 0.5 0.45 ; ; I I : I : i . - I .............- I - -------- --- - ...... 1 ...... une backbone j Ring backbone j otar backbone ! - 0.4 I I I i wandom tree backbone ■ ....... . * 0.35;- - - .... , . . . . . i . : l i i . _ .. — i— ........ 0.31- - - i : < : i ; : i ■ i i i ■ ! I 1 ! ... .. - j ---- | ........................... 1 1 i i : 1 1 1 1 0.2 I I ! J 1 > 0.15 ■ J : - ......; ” - I i I i ; i > 0.1 . . i ! i . i i _ _ J_______ .......... ■ 0.05 i ? i , i ; ; ; 1 °0 50 100 150 200 250 300 350 ~4Q 0~ 450 number of edges Figure 6.19: Comparison of E for scale free graphs of order 50. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 167 Line backbone Ring backbone Star backbone Random tree backbone 3.5 • - 3 0 3 1 J 2.5 I a 2 800 1000 number of edges Figure 6.20: Comparison of E (6q) for scale free graphs of order 100. 0.45- - 0.4 - 0.35 - - - 0.3;- - - Line backbone j Ring backbone Star backbone Random tree backbone ' 800 1000 1200 1400 1600 1800 number of edges Figure 6.21: Comparison of E for scale free graphs of order 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 168 Line backbone rang backbone Star backbone Random tree backbone e = 5 0.41 0.3 0.2 0.1 60 100 120 140 160 180 200 220 240 260 Figure 6.22: The degree probability distribution function that results from the numerical simulation of growth with uniform attachment graphs with different backbones. The simulation shows that although the star backbone has the smallest expected delta compared with the other backbones, the random tree backbone has the small est expected normalized delta in the middle range of the sizes of the graphs. The other backbones yield slightly different expected normalized delta’s. 6.4.4 Growth with uniform attachment graph The degree distribution and density functions for the growth with uniform attach ment graphs with different backbones are shown in Figures 6.22 and 6.23, respec tively, where the parameters for these generators are n — 3000, and m = 147,550 for random graphs with Ring backbone, m = 147,549 for random graphs with Line, Star, and Random tree backbones. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 169 b : - ■ - ............. ; .........I........- ■ ■ ..........;.................; Line backbone ; Ring backbone ; Star backbone i Random tree backbone i 1 ’ , ■ ■ - - y - - :......... - - - j- - - — - - - - - . . . I . . . ■ ; ! : ; : ; : : 1 ■ ■ 60 80 100 120 140 160 ' 180 200 220 ~24o’ ” 260 degree Figure 6.23: The degree probability density function that results from the numerical simulation of growth with uniform attachment graphs with different backbones. The E (5g) and E for growth with uniform attachment graphs with dif ferent backbones are shown in Figures 6.24, 6.25 for n — 50 and in Figures 6.26, 6.27 for n = 100. In contrast to the scale free graph, the random graph generated by growth with uniform attachment does not depend upon its backbone topology. The four backbones seem to provide nearly indistinguishable results for the expected delta and expected normalized delta as the size of the graph is increasing. 6.4.5 Comparison The comparisons of the E (8a) and E (gf^) among random graph generators are shown in Figures 6.28, 6.29 for n = 50 and in Figures 6.30, 6.31 for n — 100. Here, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 170 Line backbone • Ring backbone Star backbone Random tree backbone 200 250 number of edges Figure 6.24: Comparison of E (Sq) for growth with uniform attachment graphs of order 50. 0.5,...........- - 0 .45- - - - 0.4 0.25" - - - - | 0.2 r UJ 0.15'- Line backbone Ring backbone Star backbone Random tree backbone 150 200 250 number of edges Figure 6.25: Comparison of E for growth with uniform attachment graphs of order 50. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 171 1g 2.5 - Una backbone j Ring backbone ■ Star backbone j Random tree backbone : 800 1000 1200 1400 1600 1800 number of edges Figure 6.26: Comparison of E (5q) for growth with uniform attachment graphs of order 100. 0.4;- 0.35 - Une backbone Ring backbone Star backbone Random tree backbone 800 1000 1200 1400 1600 1800 number of edges Figure 6.27: Comparison of E for growth with uniform attachment graphs of order 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 172 Random graph j Small world graph ; Scale free graph i Growth with uniform attachment graph i 100 150 200 250 300 350 400 450 500 number of edges Figure 6.28: Comparison of E (Sq) for all random graph generators of order 50. the backbone graphs in random graphs, scale free graphs, and growth with uniform attachment generators are the random trees. In addition, the (5 parameter for small world graph is 0.5. The simulation suggests the following conclusions: 1. As the size of the graph is increasing, the expected delta is decreasing. In contrast, the expected normalized delta is first decreasing as the expected delta is decreasing, then approximately constant after its reaches its mini mum; finally increasing as the diameter is decreasing. This shows that the hyperbolic property occurs in the middle range of the sizes of the graphs. In the beginning, a graph is a tree (except for small world graph) where the Sq vanishes. As the size of graph is increasing, the expected delta abruptly R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 173 i °-3\ C O i tj I | 0-251- I 0.2;- 0.15) ! 0.1 i - • Random graph Small world graph Scale free graph Growth with uniform attachment graph 50 100 150 200 250 300 350 400 450 500 number of edges Figure 6.29: Comparison of E for all random graph generators of order 50. Random graph small world graph scale free graph Growth with uniform attachment graph 0 200 400 600 000 1000 1200 1400 1600 1800 2000 number of edges Figure 6.30: Comparison of E {5 a) for all random graph generators of order 100. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 174 Random graph > Small world graph Scale free graph i Growth with uniform attachment graph i 0 200 400 600 800 1000 1200 1400 1600 1800 2000 number of edges Figure 6.31: Comparison of E ( f c ) for all random graph generators of order 100. increases to a certain value and then continuously decreases. In the middle range of the sizes of the graphs, the expected delta reaches a minimum and no longer decreases. This explains the flat curve in the middle range. As the size of the graph is continuously increasing, the graph becomes more of a complete graph and the diameter is decreasing until it is comparable to the 5q- This yields an increase of the expected normalized delta. Therefore the graph is not large enough to observe the hyperbolic property. 2. Although the expected delta in the scale free generator is not less than that of the other random graph generators, the expected normalized delta for the scale free generator is the minimum among all generators. This suggests that the scale free graphs are more hyperbolic (in the sense of 5g) than the other R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 175 random graphs. 3. The expected normalized delta’s in the scale free and in the growth with uniform attachment cases have longer middle ranges than the random graphs and small world graphs. This suggests that the graphs generated from the growth process seem to provide longer range of hyperbolic properties than the graphs without growth process. 4. All random graph generators have the hyperbolic property occurring around the middle range of the sizes of the graphs. This corresponds to a graph which intuitively has the probability p of an edge between two different vertices around 0.15 — 0.25. This follows from the fact that p is roughly the fraction of the size of the graph to the total possible number of edges. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 176 Chapter 7 Application of hyperbolic geometry to communication networks In this chapter, the Internet and WWW are viewed as hyperbolic spaces and a hyperbolic geometry approach is devised for an alternate routing strategy to defeat the attack scenario in which a communication link is compromised by eavesdrop ping. Since TCP is not robust against out-of-order packet arrival, alternate routes should be restricted to be along near optimal paths. The alternate paths can be con structed as &-local geodesic paths since in hyperbolic space, the ft-local geodesics are quasi-geodesics This chapter involves the following: First, curvature of com munication networks; second, the connection between the metric graph and the geodesic space; finally, the Mocal geodesic routing algorithm. 7.1 Curvature in communication networks The internet and WWW are probably the communication networks that have most profoundly influenced research in this century. The size and complexity of these R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 177 communication networks grow so rapidly that they are not only part of many people life but also part of their future. This is probably what is driving the field of com puter science and applied mathematics towards the study of the so-called complex networks. Complex networks range from communications network to biological net works e.g., neural networks, protein folding. The complex networks are considered in their essence as graphs consisting of vertices connected by edges. Edges can be directed or undirected, physical or logical. In this section, the study of complex networks is mainly focusing on its applications to the internet and WWW. The Internet is a global communications network consisting of physical links between computer and other telecommunication devices. The internet topology can be studied at 2 levels (see [AB02] and [DM02]): the router level and the interdomain (autonomous system) level. In the router level (see Figure 7.1), the internet can be considered as a graph consisting of vertices corresponding to the routers and computers, and of edges corresponding to the physical communication links. In the interdomain level, the internet can be considered as a graph where each vertex represents a domain composed of routers and subnetworks managed by a identifiable organization and each edge represents the connection between each domain. There exists a connection between two domains iff there exists a physical path from one domain to the other, managed by the border gateway protocol, without passing through any other Autonomous System. The WWW (see Figure 7.2) is a virtual network consisting of documents (web R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 178 Figure 7.1: The topology of the Internet at router level. Figure 7.2: The topology of the WWW: @mathiab.usc.edu. pages) and of the directed hyperlinks (URL’s). The hyperlink is a logical connection between two documents. Because of the overwhelming curse of dimensionality in the internet and WWW, it is tempting to approximate the internet and WWW with a continuous geometric structure and to model the traffic by a flow described by partial differential equa tions. The internet and WWW can be viewed as graphs drawn on the surface S 2 of the earth with the possibility of handles to remove edges crossing as explained in R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 179 section 2.3. In fact, if the genus of an orientable surface is greater than 2, then it is follows from the application of Gauss Bonnet theorem (see Thm. 23) that the sur face can be endowed with a hyperbolic metric and the graph written on the surface has the possibility to be hyperbolic from the large scale. The process of removing the edges crossing does not provide a unique number of handle. Although, this lack of uniqueness of the number of handles can be removed by using the minimum genus embedding, this lead to an JVP-complete problem [Tho89]. Perhaps the most straightforward way to view the internet and WWW is to consider them as geodesic metric spaces obtained from their weighted directed graphs [JL02]. The concept of curvature is easily derived from metric curvature. Note that another concept of curvature of communication networks has been defined independently by Eckmann [EM02]. In Eckmann’s definition, the local curvature at vertex n is defined by ^ = - i r h P -1) where tn is the number of triangles containing n as a comer and vn is the number of edges leaving the vertex n. This concept of curvature is certainly related to the clustering coefficient defined by Watt and Strogatz [Wat99]. It also appears to be related to the concept of Alexandroff angles (see [BH99, Chap. 1, Def. 1 .12] and [BBI01, Chap. 4, Sec. 4.3]) at the cones of the triangles, a concept which also leads to local curvature. The internet and WWW can be viewed as geodesic metric spaces with negative R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 180 curvature, from the fact that every finite graph is a hyperbolic space. In addition, the scale free generator constructs a random graph with the power law degree distri bution [BAJ99] and is more hyperbolic than the other generators in the sense that the expected normalized delta is smaller than that of any other random graph gen erators (see section 6.4.5). This suggests a possible connection between the power law degree distribution and the hyperbolic property. In the study by Faloutsos et al. [FFF99], the internet topology has a degree distribution that follows the power law in both router and interdomain levels. In addition, in the study by Barabasi [BA99], the WWW has a power law degree distribution. This suggests the possible presence of the negative curvature in the internet and WWW. Baryshnikov [Bar02] considered the network as a metric space and argued that the internet is in fact a hyperbolic space. As an illustration, the Internet Service Provider (ISP) graph is shown in Figure 7.3. In the ISP graph, each vertex corresponds to an ISP (that is a cluster of hosts and servers managed by the same organization, with nearly matching Internet Protocol (IP) addresses) and each edge corresponds to commu nication link between ISPs via the Border Gateway Protocol (BGP). Each edge is assigned a weight equal to the number of paths observed between the two ISP’s by a traceroute-like routine. The ISP graph clearly illustrates the “core-concentric” property. In addition, the CAIDA project revealed that many other graphs, like the Autonomous Systems graphs, have the same property. As with any graph con sisting of a highly connective core and long tendrils, the geodesic lines joining three R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 181 A Figure 7.3: The ISP graph consisting of a highly connective core and long tendrils. Ob serve that the geodesic triangle, A ABC is slim. points A, B, C at the ends of the tendrils are forced to transit via the highly con nective core, contributing to the slimness of the geodesic triangle, A ABC. It turns out that the core-concentric graph is hyperbolic by nature. One of the many security concerns in communication networks is eavesdropping or packet sniffing, that is, unauthorized packet interception along a link with the potential of reconstructing the full message. As TCP does in normal condition, almost all packets are sent along the same optimum path from source to destina tion which results in an easy eavesdropping. One of the proposed patches to such a security breach is to send packets in a randomized fashion along multi differ ent paths [HB01]. This multipath routing not only hardens the network against eavesdropping but also improves resistance to traffic analysis and reliability of net R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 182 work. However the multipath routing, different paths may have different delays. The out-of-order packet arrival at the destination could create drops if the arrival sequence is altered by more than 3 slots. Unless some robustness TCP protocol is implemented, it is imperative to send the packages along nearly optimal paths having delays as close as possible to the delay in the optimal path. In hyperbolic geometry, the nearly optimal paths can be formalized as the quasi-geodesics and the quasi-geodesics remain in an identifiable neighborhood of the geodesic. In addition, a fc-local geodesic is in fact a quasi-geodesic in hyperbolic space. This suggests the implementation of multipath routing along the &-local geodesic paths. 7.2 Metric graph and Geodesic space A weight directed graph can be considered as a geodesic metric space by metrizing the individual edges of the graph as bounded intervals of the real line, and then defining the distance between two points to be the infimum of the lengths of paths joining them, where the length is measured using the chosen metric on the edges. Prom the Hopf-Rinow theorem (Thm. 27), a directed graph of finite order and of finite size is a geodesic metric space. A path p — po,... ,pk can be considered as a mapping parameterized by arc length from a closed interval [0,1 ] to a graph G where I is the path length of p. In particular, the geodesic from vertex s to vertex d is the shortest path from vertex s to vertex d with the arc length parameterization. Hence with an arc-length parameterization, the (A, e) quasi-geodesic 7 is a path R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 183 from s to d such that \ l h[tut2]) - £ < d { 7 (ti), 7 (^2)) < a/ ( 7 [tl> t3 ]) + e where t i ,t 2 £ [0, £ ] and I is the path length. That is equivalent to j d ( 7 (£1), 7 (£ 2)) - j < I (7[tl,t2 ]) - Xd ^ (*a)) + A e. Since d ( 7 (£x) , 7 (£2)) < I ( 7 [ tljt2]) , then the previous condition requires only check ing that 1 bltite]) ^ Ad (7 (h), 7 (*2)) + Ae. Intuitively, the length of the path between any two vertices cannot exceed approx imately A times the length of the corresponding geodesic. The fc-local geodesic is such that for each pair of £ 1, £2 6 [ 0 , 1 ] with |£i — £ 2| < k, is the shortest path joining 7 (£1) to 7 (£2) . A routing algorithm is an algorithm to determine the path between two vertices, i.e., the geodesic path between two vertices. Since the quasi-geodesic path is not far from the geodesic, then it can be used as an alternate path to improve security in the networks. With the property that a fc-local geodesic is a quasi-geodesic, the con struction of k-local geodesics yields quasi-geodesics. The algorithm for constructing fc-local geodesic can be considered as in the following section. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 184 7.3 R outing algorithm via fe-local geodesic In this section, the routing algorithm via fc-local geodesics is considered. The intuitive idea to construct the fc-local geodesics is to recursively concatenate length k geodesics in such a way that the resulting path is still fc-local geodesic (see Figure 7.4 ). Observe that every fc-local geodesic of length less than or equal to k is in fact a geodesic. In addition, if (u, v) is a directed edge with w (u , v) > d (u, v) > k, then the directed edge (u, v) is a fc-local geodesic. Therefore, the construction of a fc-local geodesic path requires the computation of all geodesics of length less than or equal to k. Given that pi and P2 are fc-local geodesic paths, then to ensure that the path p resulting from concatenating pi and P2 is still a fc-local geodesic, the last sub-path of length k In pi should be the same as the first sub-path of length k in p2- The fc-local property of the resulting path p follows from the fc-local geodesic property of pi and p2- It turns out that fc-local geodesic routing path can be constructed via concatenating 2fc-local geodesics as in the previous discussion. Hence all geodesics of length less than or equal to 2k need to be computed. This computation can be done by running Dijkstra’s algorithm for each vertex to construct all geodesics of length less than or equal to 2 k. Given that all Mocal geodesics have been constructed, then the vertex y in the A;-local geodesic path p = u ■ ■ ■ yz ■ ■ ■ v from vertex u to vertex v is said to be the ^-predecessor of p if d (z, v) < k and d (y, v) > k. In the case where d (u, v) < k, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 185 V2 V a V „ VS Figure 7.4: Construction A :-local geodesic via recursively concatenating fe-local geodesic (in this case k = 2). 7 rf c (u, w ) = u. The 2fc-predecessor vertex is defined in a similar way. Define the predecessor matrix II — [tt (u, u)] to be such that tt (u, v) is a mapping from the 2k- predecessor vertex to fc-predecessor vertex of a fc-local geodesic paths from vertex u to vertex v. The predecessor matrix II can be computed for each source vertex u as follows: Define T (u) to be a vector consisting of all vertices for which the fc-local geodesics from u was discovered. T (u) can be initialized from the knowledge of the geodesics of length less than or equal to k. In addition, for each vertex x with d (u, x) < 2k, 7 r (u, x) can be determined. For each vertex x with k < d(u, x) < 2k, concatenating the /c-local geodesic from u to x with all geodesics of length less than or equal to 2k that begin with the vertex in the image of 7r (u, x ) , pass through x, and end at vertex y £ T (u). Update the predecessor matrix with new paths and add new vertices to T (u ). At this step, all vertices of length less than or equal to 3k have been discovered. The algorithm repeatedly adds new vertices until all vertices have been discovered. Given the predecessor matrix II and all geodesics of length less than or equal to R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 186 k, then the /c-local geodesic path for each source vertex u and destination vertex v can be discovered as follows: Select the fc-predecessor and 2fc-predecessor vertices from 7 r (u, v), and call them x and y, respectively. The path from x to v is known since it is in fact a geodesic from x to v of length less than or equal to k. The 3k- predecessor vertex can be selected from the inverse image of y under the mapping 7r (u, x ) and the path from y to x can be constructed. Continue this procedure until the source vertex u is discovered. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 187 Chapter 8 Conclusion 8.1 Research summary In this dissertation, several concepts of hyperbolic geometry related to communi cation networks have been introduced. The various metric curvatures 5 and the refined upper bound of the Hausdorff distance between the geodesic and the quasi geodesics have been derived for a Riemannian manifold with a constant negative curvature. The scale free generator, which is used as a topology generator in com munication network, has been shown to enjoy the best hyperbolic property among several random graph generators. This result leads to the probable conclusion that a communication network can serve as a prototype of metric hyperbolic space. With this hyperbolic property, multi-path routing in communication network has been accomplished by sending packets along fc-local geodesics. It turns out that the alternative paths are sufficiently close to the optimal path. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 188 8*2 Future Perspectives There are several issues not addressed in this research that can be proposed for future research. 1. Nonsymmetric distance in communication networks: In this dissertation, the curvature of communication networks has been approached via the topolog ical aspects of random graphs. That is, there is no distinguished between two directions along an edge of the graph. However, the weight functions in communication networks need not to be the same and the communication cost from vertex A to vertex B is not in general equal to the communication cost from vertex B to vertex A, i.e., d (A, B) d (B , A ). This unsymmetrical distance function can be formalized through the concept of Finsler geometry (see [SheOl]). Finsler geometry uses the Minkowski norms, instead of inner products, to describe geometry and can be intuitively viewed as Rieman- nian geometry without the quadratic restriction in an inner product [Che96]. However, the combination of Finsler geometry and coarse geometry remains a challenging open field. 2. Worm propagation on hyperbolic networks: A worm in a communication network is a malicious piece of code that can replicate itself and propagate through a communication network by exploiting software vulnerabilities. In order to develop a worm defense mechanism, it is necessary to understand the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 189 general pattern of a worm propagation on a hyperbolic graph. However, this creates a problem of generating a great many hyperbolic graphs, so that the general pattern of the propagation of the infection could be understood. This problem can be approached by using Cayley graphs as prototype of hyperbolic graphs. The concept of worm propagation on Cayley graphs is related to the so-called growth function, defined to be the mapping that associates to every integer k > 0 the number (5(k) of elements g G F such that ds{l,g) < k where Gs (T) is the Cayley graph [dlH O O , Chap 6 , Sec. 6 A ]. The growth series B(z) and spherical growth series E (z) are defined as 00 B(z) = (8.1) k~0 cc E M = (8.2) fc = 0 where a (k) = /3(k) — (3(k — 1), er(0) = 1. These growth and spherical growth series determine how fast the worm can propagate through the Cayley graph. 3. PDE formulation of information flow in communication networks: The ob jective of this problem is to embed communication networks into continuous geometric structures such as manifolds and describe the information flows by PDE’s on manifolds. With the new PDE theory developed by Gromov [Gro8 6], the self-similar behavior of the Internet traffic signals can be ex plained. Clearly, the flow is defined on the edges. However, after interpo lating the graph with a manifold, it is necessary to define the flow on the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 190 surface elements as well. This corresponds to the problem of extension of vector fields. As is well known, there might be obstructions to extending the flow. Whether the information flow in hyperbolic network can be described by PDE approach is still an open problem. 4. Robustness of against curvature uncertainty: In communication networks, the weight function defines the metric d, which in turn defines the curvature. This weight function is in fact a time dependent function, i.e., it can be changed over the time during the operation of the communication network due to some uncertain factors such as congestion recently observed over the link, delay, packet error rate, etc. It turns out that the stability of the geodesics and quasi-geodesic under this feedback scheme is still questionable. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 191 [AB02] [BA99] [BAJ99] [Bar02] [BBI01] [Bea83] [BH99] [BolOl] [Boo02] [BR] Bibliography R. Albert and A.-L. Barabasi. Statistical mechanics of complex net works. Reviews of Modem Physics, 74:47-97, January 2002. A.-L. Barabasi and R. Albert. Emergence of scaling in random net works. Science, 286:509-512, 1999. A.-L. Barabasi, R. Albert, and H. Jeong. Mean-field theory for scale- free random networks. Physica A, 272:173-187, 1999. Y. Baryshnikov. On the curvature of the internet. In Workshop on Stochastic Geometry and Teletraffic, Eindhoven, The Netherlands, April 2002. D. Bur ago, Y. Burago, and S. Ivanov. A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. American Mathemat ical Society, Providence, Rhode Island, 2001. A. F. Beardon. The Geometry of Discrete Groups, volume 91 of Grad uate Texts in Mathematics. Springer-Verlag New York Inc., New York, New York, 1983. M. R. Bridson and A. Habfliger. Metric Spaces of Non-Positive Curva ture, volume 319 of A Series of Comprehensive Studies in Mathematics. Springer-Verlag, Berlin-Heidelberg, 1999. B. Bollobds. Random Graphs. Cambridge Studies in Advance Math ematics. 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Birkhauser, Boston, Massachusetts, 1999. J. L. Gross and T. W. Tucker. Topological Graph Theory. Dover Pub lications, Mineola, New York, 2001. J. P. Hespanha and S. Bohacek. Preliminary results in routing games. In Proc. Of the 2001 American Control Conference, June, 2001. M. Henle. Modem Geometries Non-Euclidean, Projective, and Dis crete. Prentice-Hall, Inc., Upper Saddle River, New Jersey, second edition, 2 0 0 1 . C. J. Isham. Modern Differential Geometry for Physicists, volume 61 of World Scientific Lecture Notes in Physics. World Scientific Publishing Co. Pte. Ltd., River Edge, New Jersey, 1999. E. A. Jonckheere and P. Lohsoonthom. A hyperbolic geometry ap proach to multi-path routing. In Proceedings of the 10th Mediterranean Conference on Control and Automation (MED 2002), Lisbon, Portu gal, July 2002. FA5-1. J. Jost and X. Li-Jost. Calculus of Variations, volume 64 of Cam bridge Studies in Advance Mathematics. Cambridge University Press, Cambridge, United Kingdom, 1998. J. 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Hyperbolic geometry of networks
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