University of Southern California Dissertations and Theses

Traffic pattern in negatively curved network

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Traffic pattern in negatively curved network

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Content TRAFFIC PATTERN IN NEGATIVELY CURVED NETWORK

by

Mingji Lou

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)

December 2008

Copyright 2008 Mingji Lou

ii
Epigraph
“There is nothing either good or bad, but thinking makes it so.” Hamlet, William.
Shakespeare; Act II, scene ii.

iii
Dedication
To my parents.

iv
Acknowledgments
There have been many hands helping me in my work, and some people stand out in my
mind. The first idea related to this thesis came from my course project, Tracking and
Mitigating Piracy Propagation in Network, in Professor Edmond Jonckheere’s course
Networking Control. Professor Jonckheere encouraged and advised me through the whole
journey of my Ph.D. studying including this thesis, our other projects and patents. Dr.
Yuliy Baryshnikov from Bell Lab introduced me to the interesting traffic congestion
problem in networks and provided me with several bench mark cases during his visit to
USC and my summer visit to Bell Labs. Professor Francis Bonahon from the Department
of Mathematics always helped me understand and overcome the most difficult
mathematical concepts and proofs. Professor Bhaskar Krishnamachari guided me to
interconnect the theory and real-world networking applications, such as the load-balancing
problem, etc.
Many other distinguished Professors helped me during the many different periods of
my academic career; these include Professor Petros Ioannou and Professor Michael
Safonov from the control group, Professor Melvin Breuer, Professor Jerry Mendel,
Professor Gandhi Puvvada, and Professor Robert Scholtz from computer engineer and
signal processing groups, respectively, and Professor Yuming Zhang from University of
Kentucky.
The research group members, Fariba Ariaei and Srideep Musuvathy, have been a great
resource for my work.
I also want to thank Mr. James S. Turner and the Global Gateway Foundation for

v
supporting another academic research project of mine, Spinal Injury Imaging by
Magnetically Levitated Sensors.
If the meaning of pursuing a Ph.D. is more than a degree, all of the lessons I learned
from this journey has permeated my whole life. I feel profoundly enriched and fortunate to
have met all of those whom I interacted with.

vi
Table of Contents
Epigraph ii

Dedication iii

Acknowledgments iv

List of Tables viii

List of Figures ix

Abstract xiii

Chapter 1: Introduction 1

Chapter 2: Graph Curvature by Local Measurement 10
2.1 Alexandrov Angles 11
2.2 Clustering Coefficient 15
2.3 Instances of Negatively Curved Graphs 17

Chapter 3: Large Scale Gromov Curvature 20

Chapter 4: Mathematical Model and Conjecture 23

Chapter 5: Benchmark Examples 28
5.1 Simulation Results of Negatively Curved Graph 28
5.1.1 Simulation Setup 29
5.1.2 Results: Conjecture G- 30
5.1.3 Results: Conjecture G 31
5.1.4 Non-symmetrical Negatively Curved Networks 33
5.2 Vanishing Curvature—Planar Graph of Valence 6 35
5.3 Nonpositive Curvature 36
5.3.1 Planar Graph with Random Valence 6, 7, or 8 36
5.3.2 Comparison I: Between Planar Graphs of Vanishing,
Negative and Nonpositive Curvature 39
5.3.3 Comparison II: Between Planar Graphs of Vanishing
and Negative Curvature 40
5.4 Positive Curvature 42
5.5 Dijkstra’s Algorithm with Random Pick-up 43
5.6 Regular Binary Tree 47
5.6.1 Basic Definitions 47
5.6.2 Traffic Load on Top of Tree 49
5.6.3 Recursion on Traffic 49

vii
5.6.4 Results 51

Chapter 6: Mathematical Proofs 54
6.1 Proof Conjecture G+: 53
6.2 Proof Conjecture M- 55
6.3 Quantitative Measure of Conjecture ± G in Disk 58
6.3.1 Traffic at the center of a Euclidean Disk, + G: 58
6.3.2 Conjecture − G 61

Chapter 7: Shortest-path Routing Vs. Load Balancing Routing
in Networks of Different Curvature 71

Chapter 8: Summary and Future Works 83

References 84

Appendix 87
1 Tracking and Mitigating Piracy 87
2 Optimize Parallel Computing Performance 91
2.1 A Case for Positive Curvature 94
2.2 Partition and Combination 97
2.2.1 Heuristic Methods 99
2.2.2 Hilbert Cube Method 101
2.2.3 Comparing the Partitioning Results 103
2.3 Some Further Speculations 103
3 Localization and Synchronization in Sensor Network 104
4 Waterflooding in Petroleum Engineering 108

viii
List of Tables
Table 5-1: Dijkstra’s Algorithm with Random Pickup among the
Multiple Minimum Paths Instead of First Minimum Path Pickup 44

Table 7-1: In Load Balancing, the Asterisk (*) Keeps Rotating
among the Equal Cast Paths 79

ix
List of Figures
Figure 1-1: The Traffic Congestion in the Communication Network 2

Figure 1-2: Waterflood in Petroleum Engineering 3

Figure 1-3: The Tradition Understanding about Congestion: Congestion
Occurs at “Core” 7

Figure 1-4: A Hybrid Network: A Network That Has a Small-World
(Positive Curvature) and a Scale-Free (Negative Curvature)
Component Connected by a Bidirectional Cross-Feeding Mechanism 8

Figure 2-1: Delaunay Triangulation of Nonuniformly Distributed Agents 18

Figure 2-2: Clustering Coefficient Distribution of Delaunay Triangulation 18

Figure 2-3: Alexandrov Angles Distribution of Delaunay Triangulation 19

Figure 4-1: Assume the Demand Is Uniform, 1 ) , ( = Λ t s
d
and the Routing is
Optimal, the Traffic Load in U is 5 While the Blocking Measure is 4 24

Figure 5-1: The Simple Hyperbolic Graph with Node Degree Seven
and Uniform Edge Length 29

Figure 5-2: The Traffic Load Distribution of Nodes in a Hyperbolic Network
of Valence 7 with 100 Nodes under Random Pickup Routing 31

Figure 5-3: Traffic and Inertia in the Simple Hyperbolic Graph with Node Degree 7 32

Figure 5-4: Traffic and Inertia in the Simple Hyperbolic Graph with Node Degree 9 33

Figure 5-5: The Traffic and Inertia Distribution in an Unsymmetrical Network
(Vertext#3 Has Maximum Traffic and Minimum Inertia) 34

Figure 5-6: The Traffic and Inertia Distribution in an Unsymmetrical Network
(Vertext#3 Has Maximum Traffic and Minimum Inertia) 35

Figure 5-7: The Traffic Load Distribution in the Euclidean Plane with Greedy
Random Pickup Routing. 36

Figure 5-8a: The Traffic Distribution in the Planar Graph with Random Node
Degree (6, 7 or 8). 37

x
Figure 5-8b: The Traffic Distribution and the Inertia Distribution (Vertex#1 Has
the Minimum Inertia and Maximum Congestion) 38

Figure 5-8c: The Traffic Distribution and the Inertia Distribution (Vertex#1 Has
the Minimum Inertia and Vertex#4 Has Maximum Congestion) 39

Figure 5-9: The Traffic Distribution in Different Planar Networks, Degree 6
Everywhere, Degree 7 Everywhere, Degree 8 Everywhere, and
Random degree 6/7/8; The Routing Algorithm Implements Random
Pickup 40

Figure 5-10: The Maximum Traffic and the Minimum Inertia versus the Node
Degree of a Uniform Graph 41

Figure 5-11: The Traffic in the Congestion Center, the Total Traffic in the
Network and the Total Connection in the Network 41

Figure 5-12: Positively Curved Network (Node Degree=5, Icosahedron)
without Random Pickup. 44

Figure 5-13: Positively Curved Network (Node Degree=5, Icosahedron) with
Random Pickup among the Multiple Minimum Paths and 10 Trial
Average 45

Figure 5-14: Two Geodesics Connecting a Pair of Vertexes. (Red Circles:
the Vertexes along One Geodesic, Blue Circles: along the
Other Geodesic) 46

Figure 5-15: The Congestion Mitigation Resulting from Randomization of the Pickup 46

Figure 5-16: The Diagram of Traffic among Layers 51

Figure 5-17: The Traffic Distribution by Using Dijkstra Algorithm and Simulation 52

Figure 5-18: The Traffic Load for the Nodes in Different Layer by Analytical
Calculation 53

Figure 6-1: Traffic in Euclidean Disk 59

Figure 6-2: Traffic Model in Hyperbolic Disk 61

Figure 7-1: A Typical Packet Multicasting in Different Layers 72

Figure 7-2: Two Methods of Constructing Multicast Algorithms: (a) Dense Mode,
Using a Source-based Tree; (b) Sparse-mode, Using a Shared Tree 72

xi
Figure 7-3: A Snapshot of the Visualization of Network with Node Degree 6 by
Using the Nam of NS-2 73

Figure 7-4: A Snapshot of the Visualization of Network with Node Degree 8 and
Its Packet Loss by Using NS-2 Nam 74

Figure 7-5: The Queue Size of the Duplex-link between Node #0 and #1 in Node
Degree 8 Network 75

Figure 7-6: The Link Utilization of the Duplex-link between Node #0 and #1 in
Node Degree 8 Network 75

Figure 7-7: The Packets Loss of the Duplex-link Between Nodes #0 and #1 in
Node Degree 8 Network 77

Figure 7-8: The Queue Size of the Duplex-link Between Node #0 and #1 in
Node Degree 6 Network 77

Figure 7-9: The Link Utilization of the Duplex-link between Node #0 and #1 in
Node Degree 6 Network 78

Figure 7-10: The Packets Loss of the Duplex-link between Node #0 and #1 in
Node Degree 6 Network 78

Figure 7-11: The System Diagram of Curvature Based Load-balancing 81

Figure 7-12: Traffic Distribution of Node Degree 7 Network (Left: without
Load-balancing; Right: with Load-balancing) 82

Figure 7-13: Routing with and without the Curvature Based Load Balancing
(Red Circle without Load Balancing, Blue Circle with Load Balancing) 82

Figure A-1: Piracy Propagation Model in the Decentralized Small-world Network
(Left) and Centralized Scale-free Network (Right) 88

Figure A-2: Left, Distribution of Theater/Internet Release Time Lags for Samples
in AT&T Data Set. Week 0 is the Week a Movie was First Released
in U.S. Theaters. Right: Simulation Results of a Piracy Index Increment
Distribution 89

Figure A-3: The Joint Graph with Scale Free and Small World Graphs Is Isolated
by Cheeger Isoperimetric Bi-partition. (Red nodes Are Belong to the
Small World Graph; and the Blue Nodes Are the Vertexes of the Scale
Free Graph) 90

xii
Figure A-4: A Typical Networked Parallel Computer “Grid” 92

Figure A-5: An Example of Parallel “Hello World” Program with the Instruction
Flow, from: http://www.llnl.gov/computing/tutorials/parallel_comp/ 92

Figure A-6a: Hierarchy View of a 802.11 Receiver Module with a Viterbi Decoder
in Xilinx FPGA 93

Figure A-6b: Initial Placement of a 802.11 Receiver Module with a Viterbi Decoder
in Xilinx FPGA 93

Figure A-7: Cheeger Constant Evolution of the Sub-graphs in Recursive
Bi-partitioning (the Order of the Original Graph is 1000) 98

Figure A-8a: (by Hilbert Cube Method) 1st Bi-Partition, Cut the Hybrid Piracy
Network into 2 Parts; the Cut is Consistent with the Decomposition
of the Network in the Scale-free (Blue) and the Small-world (Green)
Subnetworks 98

Figure A-8b: (by Hilbert Cube Method) 2
nd
Bi-Partition, Do a Cheeger
Partitioning of the Two Sub-networks Resulting from the First Cheeger
bipartitioning; This Leads to a Decomposition into 4 Parts: the Previous
Scale-free Part is Decomposed into Blue and Green Sub-networks and the
Previous Small-world part Is Decomposed into a Red and a Cyanic Blue
Sub-networks 99

Figure A-9: Right, the Average of the Alexandrov Curvature as Function of
Threshold; Left, Average of Clustering Coefficient as a Function of
Threshold for Data Set 06A 99

Figure A-10: Traffic Analysis for the Sensor Networks with Threshold 0.1
(Blue Line) and Threshold 0.5 (--Black Line) 107

Figure A-11: Inertia of the Sensor Networks with Threshold 0.1 (Blue Line)
and Threshold 0.5 (--Black Line) 107

Figure A-12: A Idea Lagged Waterflood Response 110

Figure A-13: Composite Spider Diagram for Oil Responses. 110

Figure A-14: An Idea Waterflooding in the Reservoir (Left: Before the
Waterflooding; Right: after the Waterflooding; Blue Represents
Water and White Represents Oil) 111

xiii
Abstract
In this thesis, the fundamental experimental observation is that, on the Internet and
other networks, traffic seems to concentrate quite heavily on some very small subsets. The
main result in here is that this phenomenon is not, in general, related to the popular “heavy-
tailed” phenomenon, but is a consequence of the negative curvature of the network. The
mathematical analysis and simulation results confirm this striking traffic pattern specific to
negatively curved networks from both the theoretical and practical points of view.
Furthermore, this thesis addresses another fundamental question: if congestion does not
necessarily occur at vertices with high degree, nor at the so-called highly connected “core,”
then what are the congestion points? It is shown that the congestion points are the points
relative to which the network has low moment of inertia. That single point relative to which
the network has a well-defined minimum is referred to as the centroid, the point through
which most of the traffic transits. Probably the most important result as far as protocol
design is concerned is that load balancing can be achieved by a routing table designed on
the basis of a virtual network in which the link weights have been adjusted to correct the
curvature from negative to positive. The latter mathematical technique is reminiscent of the
Yamabe flow and the Poincaré conjecture. The practical implementation of this concept on
the ns-2 network simulator indicates a significant reduction of the congestion.

1
Chapter 1: Introduction
One of the most important challenges in networking systems, especially in large and
wide area networks, is the traffic congestion problem. By the definition in the Wikipedia
encyclopedia [Wikipedia08], the congestion is a state of excessive accumulation or
overfilling or overcrowding and is broadly accepted across all users in the various fields,
even though in some specific contexts, the word “congestion” receives a more precise
definition and refers to a more precise situation. For example, in computer communication
networks, congestion represents an overloaded condition when a link or node is carrying so
much data that its quality of service deteriorates as shown in Figure 1-1; in medicine and
pathology, the term congestion is used to describe excessive accumulation of blood or other
fluid in a particular part of the body; in transportation, congestion happens when the
volume of traffic exceeds the capacity of the highway; in the supply chain area, congestion
occurs when there is an accumulation of goods in the inventories, with the potential of
“deflation;” in electrical power transmission (power grid), congestion means thermal or
other breakdown of high voltage power lines resulting in the inability to supply an area
with electricity. So congestion, in most cases, is the worst state of a network and as such it
carries a negative connotation.
However, in several special situations, the system design is intended to generate
congestion, so that the latter acquires a positive connotation and is referred to as
congregation in a network. For example, the waterflood in petroleum engineering, as
shown in Figure 1-2, is a method of secondary recovery in which water is injected into the
reservoir formation to displace residual oil. The water from injection wells physically

2
sweeps the displaced oil to adjacent production wells. In this problem, an optimal control
method is used to maximize the congestion/congregation around the production wells by
controlling the injecting water or other fluids through the injector wells in the field.

Figure 1-1: The Traffic Congestion in the Communication Network

3

Figure 1-2: Waterflood in Petroleum Engineering
In this thesis, the generic communication network architecture is used as a bench mark
case to study the traffic pattern, especially the congestion pattern, but the general results
can also be applied to those other networks mentioned above. The congestion in the
communication network can be either logical or physical as shown in Figure 1-1. The
queuing feature in two routers can create a logic bottleneck between user A and user B.
Correspondingly, insufficient bandwidth on physical links between routers and network can
also create congestion. The current congestion control technologies in communication
networks are based on the feedback from the congested node to slow down the packet flow
rate from the source, such as bidirectional congestion control and random early detection
(RED). However, these passive technologies can only be applied once the congestion has
happened to some degree, and it is only based on the local point of view of some queue

4
overflow along the source to target path. This thesis will study the deeper reason behind the
congestion in the large scale, and will challenge the current least-cost-path algorithms, such
as Dijkstra’s shortest path algorithm. It will indeed be established that these least-cost-path
algorithms aggravate the congestion, especially with in negatively curved network graphs.
A fundamental ingredient in this thesis is that, in order to get a large-scale view on the
congestion problem, we utilize the coarse approximation of a network graph by a
Riemannian manifold. A graph as a mathematical idealization of a network is completely
different than a Riemannian manifold; however, the recent development of the so-called
coarse geometry under the leadership of Mikhael Gromov has given the two mathematical
structures—graphs and manifolds—the unifying framework of geodesic spaces. As a
corollary, the concept of curvature has become applicable to graphs [Ghys90, Gromov87,
Gromov99]. The fundamental mathematical idea behind this unification is to realize that
the traditional Riemannian curvature, which relies on the differentiable structure of the
manifold, can be rephrased in terms of the more primitive concept of distance [Burago01,
Blumenthal53]. Since a communication network can be endowed with a distance, which
represents the communication cost, delay, etc., its curvature can be defined. The positively
curved versus negatively curved network dichotomy roughly corresponds to the more
traditional meshed (decentralized) versus core-concentric (centralized) network dichotomy
[Lou06]. One extremely important point that will be made in this thesis is that, contrary to
traditional believe, congestion is not necessarily a manifestation of the heavy-tailed
behavior related to the node degree, but is a manifestation of a more subtle process that can
be traced back to the curvature. It is argued that the concept of network curvature is about
to become a new paradigm, as demonstrated by the following early developments:

5
1) As shown by Baryshnikov [Baryshnikov02], measurements on the Internet indicate
that it is negatively curved in the sense of Gromov.
2) As shown by Eckmann and Moses [Eckmann02], the curvature of the co-links
reveals hidden thematic layers in the World Wide Web.
3) As shown by Jonckheere [Jonckheere02, Jonckheere04, Jonckheere07a,
Jonckheere07b], the celebrated growth/preferential attachment process as a mean
to construct a scale-free graph in fact leads to a (scaled) Gromov negatively curved
graph.
4) A Gromov negatively curved network has a great many quasi-optimal paths in an
identifiable neighborhood of the optimum path and hence lends itself to secure
multi-path routing [Jonckheere02, Jonckheere07c].
5) The geographical localization error in a negatively curved sensor network grows
linearly with the distance from the reference sensor [Jonckheere07d].
6) The coordination of autonomous agents to detect a chemical source is most easily
accomplished by cooperatively enforcing positive curvature of the network of
agents [Ariaei07].
7) The Delaunay triangulation of nonuniformly distributed agents is negatively
curved [Jonckheere07d].
8) A peer-to-peer network is positively curved and combined with a centralized
(hence negatively curved) searching engine, it becomes a hybrid curvature network,
which has been utilized for piracy modeling [Lou06].
9) The greedy geographical routing of Kleinberg is based on embedding the network
graph in the negatively curved Poincaré disk [Kleinberg07]. This embedding is

6
accomplished with minimal distortion if the graph is negatively curved to begin
with.
10) Gromov negatively curved networks have the property that the variation of
minimum communication cost paths under variation of the link cost can be
bounded [Jonckheere02]. Probably the most sizable difference between negatively
and positively curved networks is that the latter property does not hold in positive
curvature. In a positively curved network, an arbitrarily small variation of the link
costs can result in minimum cost path variations extending all over the network.
In chapter 2, more details about the curvature measurement will be presented. From the
above citations, we can conclude that a negatively curved network is the right model for the
Internet and the growth/preferential attachment paradigm (citation #3), and that a positively
curved network is the right model for the World Wide Web application (citation #2).
It has been experimentally observed that, on the Internet and other networks, traffic
seems to concentrate quite heavily on some very small subsets. As shown in Figure 1-3,
congestion could occur at the “core” through an easy mechanism. But this thesis shows that
there are other more subtle mechanisms creating congestion. Our conjecture is that this
phenomenon is NOT, in general, related to the popular “heavy-tailed” phenomenon, but is
a consequence of the negative curvature of the network. It is believed that the negatively
curved network paradigm applies to a great many real networks. The mathematical proofs
and simulations will unveil this striking traffic pattern in negatively curved networks from
both theoretical and practical points of view. In further, this thesis studies another
fundamental question: if congestion does not necessarily occur at vertices of high degree,
nor at the so-called highly connected “core,” then what are the congestion points? This

7
thesis shows that the congestion points are related to the inertia of the network.
Although the ultimate research objective related to this topic is to have a general theory
for hybrid networks with both negatively and positively curved sub-networks combined
together (such as in terrorists’ networks to fool detection), this dissertation focuses on the
traffic pattern in negatively curved networks only. In [Lou06], we have proposed a hybrid
network by combining both Small-world social network (with positive curvature) and
Scale-free Internet (with negative curvature), as shown in Figure 1-4. At the end of this
dissertation, we will propose several ideas and concepts specific to problems in hybrid
networks, such as partitioning of the network into sub-graphs with negative or positive
curvature network only, and the Laplacian spectrum analysis of such a network.

Figure 1-3: The Traditional Understanding of Congestion: Congestion Occurs at a “Core”

8

Figure 1-4: A Hybrid Network: A Network That Has a Small-World (Positive Curvature)
and a Scale-Free (Negative Curvature) Component Connected by A Bidirectional Cross-
Feeding Mechanism.
The organization of the thesis is the following: Chapter 2 briefly presents elementary
graph curvature via simple measurement. In further, Chapter 3 discusses large scale
Gromov curvature. Chapter 4 proposes a mathematical model of network traffic and
formulates two meta-conjectures. In Chapter 5, we develop several bench mark examples
that confirm the conjectures on traffic pattern in networks of different curvatures. The
mathematical proofs of the various traffic congestion conjectures are presented in Chapter 6.
In Chapter 7, a curvature-based load-balancing routing is proposed, and its performance is
compared with a shortest-path routing in multicast communication network. The results
show that the curvature-based load balancing algorithm mitigates the traffic congestion
typical of the traditional shortest path algorithm. Finally, Chapter 8 encompasses
conclusions and discussion. Several broader applications of using the results of this
dissertation are briefly presented in the appendix, including the piracy propagation problem

9
in hybrid network, the parallel computation problem and its application in VLSI/FPGA
connection problem, sensor networking problem, and waterflooding in petroleum
engineering.

10
Chapter 2: Graph Curvature by Local Measurement
Among the applications cited in Chapter one, #1, #3, #4, #5, #9 and #10 rely on the
concept of Gromov negatively curved graphs, which is a large scale concept. The
remaining applications, #2, #6, and to a lesser extent #7 and #8, rely on the clustering
coefficient, which in some intuitive sense is a local curvature concept at the scale of the
immediate neighbors of the vertices. The clustering coefficient at a vertex a takes values
between 0 and 1 to indicate how many triangles are made up with edges flowing out of the
vertex a and it can be argued that a clustering coefficient of 0 means negative curvature
while a clustering coefficient close to 1 means positive curvature. The clustering coefficient
is easy to compute, but, as we shall see, it provides only a rough estimate of the curvature.
A first step in this dissertation is to develop a more precise local curvature concept based
on Alexandrov angles [Burago01, Jost97, Schoen84]. One such concept, the Higuchi local
combinatorial curvature at a vertex of a planar graph [Higuchi01], has already been
developed and has recently been extended to those nonplanar graphs that are 2-cell
embedded in a surface [DeVos07]. The specific contribution here is to adapt this concept to
those nonplanar graphs that are neither triangulations of surfaces nor 1-skeleta of simplicial
decompositions of manifolds. As such, while the local combinatorial curvature of a graph
reproduces the Gauss curvature of a surface, here, because of the higher dimension and the
“messy” nature of the data, we can only reproduce the weaker Ricci and scalar curvatures
of manifolds by averaging some sectional curvature for the graph at a vertex, where the
average is relative to all “sections” through the vertex a .
While the emphasis will be on the concept of local scalar curvature defined via the

11
Alexandrov angles, we will develop the clustering coefficient approach in parallel, as it has
become some standard in the industry [Eckmann02]. We will show that the Alexandrov
angle scalar curvature is consistent with the clustering coefficient approach.
The specific application that is targeted by the curvature analysis is congestion.
Congestion points can be viewed as geodesics densely packed around negative curvature
points. Negatively curved metric spaces have the “good” property that its geodesics are
insensitive to environmental changes, but in the networking context this property turns up
to be “bad.” Indeed, since in negative curvature the geodesics are insensitive to
environmental changes, the congestion points cannot be removed by small readjustment of
the link weights [Jonckheere04]. The congestion can only be mitigated at the expense of a
drastic change of the link weight to create a positively curved “virtual network.” Routing
over the latter network, mapped to the original one, yields a congestion aware routing (see
Chapter 7).
2.1 Alexandrov Angles
The difference between positive, vanishing, and negative curvature can easily be
understood by formalizing the intuitive difference between the 2-dimensional unit sphere,
the plane, and the hyperboloid. Consider a point a on the unit sphere and the locus of points
b such that cst b a d = ) , ( , where
2
0
π
< < cst and the distance ) , ( ⋅ ⋅ d is measured along arcs
of great circles. Pick a set of consecutive points
1 1 2 1
, ,..., , b b b b b
n n
=
+
on the locus and
consider the spherical triangle
1 + k k
b ab . Redraw this triangle isometrically in Euclidean
plane,
1 + k k
b b a . The vertices written with an overbar are the comparison vertices, meaning

12
that ) , ( ) , (
k k
b a d b a d = , ) , ( ) , (
k k
b a d b a d = , ) , ( ) , (
1 1 + +
=
k k k k
b b d b b d , where ) , ( ⋅ ⋅ d denotes
the distance measured in the Euclidean plane.
1 + k k
b b a is called comparison triangle.
Define the Alexandrov angle
k
α at the vertex a as
) , ( ) , ( 2
) , ( ) , ( ) , (
cos
1
2
1
2
1
2
1
+
+ + −
− +
=
k k
k k k k
k
b a d b a d
b b d b a d b a d
α . The latter is obviously derived from the
cosine law of ordinary rectilinear trigonometry [Burago01]. If, however, we bring spherical
trigonometry in the picture, some algebraic manipulations yield
) , ( sin ) , ( sin
) , ( cos ) , ( cos ) , ( cos
) , ( ) , ( 2
) , ( ) , ( ) , (
1
1 1
1
2
1
2
1
2
+
+ +
+
+ +
−
≤
− +
k k
k k k k
k k
k k k k
b a d b a d
b a d b a d b b d
b a d b a d
b b d b a d b a d

By the spherical law of cosines, the right hand side is the cosine of
k
α , the angle
1 + k k
ab b as
measured on the sphere. From the above, we get
k k
α α ≥ , and taking the sum over k yields
∑
<
k
k
π α 2 , since on the sphere
∑
=
k
k
π α 2 . Therefore,
∑
> −
k
k
0 2 α π . Next, if we redo
exactly the same on the hyperboloid, and if we compare the hyperbolic and the rectilinear
laws of cosine, we will find
∑
< −
k
k
0 2 α π . Clearly, ⎟
⎠
⎞
⎜
⎝
⎛
−
∑
k
k
α π 2 sign is the curvature
sign. The curvature concept based on the mere sign is sometimes referred to as
combinatorial curvature.
It should be geometrically clear that, as ) , ( b a d decreases to zero, so does
∑
−
k
k
α π2.
Hence for
∑
−
k
k
α π 2 to keep some local sense, it is necessary to normalize it by the area
and define the curvature at a as

13
∑
∑
+
→
−
=
k
k k
k
k
b b a A
a
) (
2
lim ) (
1
0 cst
α π
κ
, where ) (
1 + k k
b b a A denotes the area of the triangle
1 + k k
b b a . The latter can be recovered
from the metric data. Indeed, if we define the semi-perimeter
() ) , ( ) , ( ) , (
2
1
1 1 + +
+ + =
k k k k
b a d b a d b b d s , we have
() ( ) ( ) ) , ( ) , ( ) , ( ) (
1 1 1 + + +
− − − =
k k k k k k
b a d s b a d s b b d s s b b a A
Next, we have to give some sense to the preceding in the context of graphs. The
guiding idea is to think the graph as isometrically embedded in an N-dimensional space, in
which case a cyclic ordering of a subset { } { }
n k k k
ab ab ab ab ab ab
m
,..., , ,..., ,
2 1
2 1
⊆ of all edges
flowing out of the vertex a can be thought of as a surface σ . In the extreme case when
n m = , a cyclic ordering of all edges makes up a surface and the whole graph is embedded
in a surface [Mohar01]. It is, however, preferable to embed the graph is a space of higher
dimension [Blumenthal53], as this leaves more room to maneuver to make the embedding
isometric, that is, to make the embedding preserve the distance. This forces us to choose the
other extreme, that is, to think a surface to be a cyclic ordering of a minimum number of
edges. This minimum number is clearly m=3. In this extreme case, a permutation of the
ordering of the vertices is no more than a reversal of the ordering. Hence a subset of three
edges { } σ =
3 2 1
, ,
k k k
ab ab ab , disregarding ordering, makes a section σ . Hence the
combinatorial sectional curvature at the vertex a and relative to the section σ is defined as
∑
∈
− =
σ
α π σ
i
k
i
ab
k
a K 2 ) , ( , where the Alexandrov angles are defined as before. We call the

14
above combinatorial curvature as opposed to curvature. The motivation for this concept is
that it obviates the need for some “area data,” which is a bit artificial in a graph.
The above concept is consistent with the so-called Higuchi combinatorial local
curvature of a planar graph at a vertex a defined to be
∑
− −
k k
a
| |
1
2
) deg(
1
φ
, where
) deg(a is the degree of the vertex,
k
φ is the face at the vertex a comprising the edges
1
,
+ k k
ab ab , and | |
k
φ denotes the number of vertices of the face [Higuchi01]. Multiplying
this formula by π 2 and assuming that | |
k
φ is a regular polygon we get
()
∑ ∑ ∑
− = − − =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− −
k
k
k
k
k k
e α π π π
φ
π
π π 2 2
| |
2
2
, where
k
e is the external angle at a vertex of
k
φ . The connection between the two
curvature formulas is clear; however, here, the definition is extended to non-planar graphs,
independently of a 2-cell embedding in a surface.
Recall that the Ricci curvature at the point a along the direction ab is the average of
the sectional curvatures along sections passing through the direction ab and the scalar
curvature is the average of the Ricci curvature for all directions. In the combinatorial
context of graphs, we can go directly to the scalar curvature by averaging over all sections.
Averaging the combinatorial sectional curvature ) , ( σ a K makes sense only if it is properly
scaled by the area. As such, we define the local scalar curvature at the vertex a by
∑
∑
∑
⊃
=
=
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
a
i
k k
i
k
i i
i
b b a A
a
a
σ
α π
κ
3
1
3
1
) (
2
3
) deg(
1
) (
1

15
, where { }
3 1
, ,
k k k
ab ab ab
k
= σ is an arbitrary section passing through the vertex a and
) deg(a is the degree of the vertex a, that is, the number of edges abutting the vertex a.
It is important to observe that a triangle, say,
1 + k k
b ab is geodesic in the sense that the
vertices are connected by paths of smallest length. As such, if there are no paths connecting
k
b
to
1 + k
b
of a length smaller than
) , ( ) , (
1 +
+
k k
b a d b a d
, then the path
1 + k k
b b
becomes the
union of the edges
k
ab
and
1 + k
ab
to preserve the triangle inequality, and the triangle
becomes “flat” with vanishing area, which creates the potential problem of “dividing by
zero” in the above formula. We take two different ways to cope with this difficulty.
A first approach is to set the area of a negative section (
0 ) ( 2
3 2 1
< + + − α α α π
) far
smaller than the area of a positive section (
0 ) ( 2
3 2 1
> + + − α α α π
).
In the second approach, we assign a very small area, say
6
10
−
, to the nominally
vanishing area triangles in order to avoid “dividing by zero.”
2.2 Clustering Coefficient
The clustering coefficient is a purely combinatorial concept, in the sense that it relies
only on whether two vertices are connected by an edge, independently of any metric data
quantifying the strength of the connection, if any, between two vertices. Of course, in a
(undirected) weighted graph, a link may be declared to exist if its weight is less than a
certain threshold. Having done so, the clustering coefficient at node a is defined as
{}
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∈
=
+ + +
2
) deg(
, , : #
) (
1 1 1
a
E b b ab ab b ab
a c
k k k k k k

16
The numerator is the number of existing triangles with a vertex at node a, that is, the
number of triples ()
1 1
, ,
+ + k k k k
b b ab ab , where
1
,
+ k k
ab ab are two edges flowing out of a and
1 + k k
b b is a direct link joining
k
b to
1 + k
b . The denominator is the maximum number of
triangles that could possibly have a vertex at a.
It turns out that the clustering coefficient provides an approximation of the scalar
curvature in the sense that 0 ) ( = a c implies that the scalar curvature at the vertex a is
negative, while 1 ) ( = a c means that the scalar curvature is positive, with
2
1
) ( = a c the
borderline case of vanishing scalar curvature.
Indeed, take 0 ) ( = a c . This means that, for every section {}
3 2 1
, , ab ab ab = σ ,
2 ) , ( =
j i
b b d . Indeed, since
i
b and
j
b are not directly connected, any path not passing
through a has at least 2 edges; hence the length of this path is at least 2; on the other hand,
the length of the path passing through a is exactly 2; since the distance ) , (
j i
b b d is the
minimum of the lengths of the paths joining
i
b to
j
b, 2 ) , ( =
j i
b b d . This along with
1 ) , ( =
i
b a d yields the Alexandrov angle deg 180 =
k
α . Hence 0 2
3
1
< − = −
∑
=
π α π
k
k
with
the same sign for the average scalar curvature.
Now take . 1 ) ( = a c This means that for every section { }
3 2 1
, , ab ab ab = σ , 1 ) , ( =
j i
b b d .
This along with 1 ) , ( =
i
b a d yields the Alexandrov angle deg 60 =
k
α . Hence
0 2
3
1
> = −
∑
=
π α π
k
k
, with the same sign for the average scalar curvature.

17
It turns out that the “Euclidean threshold” is
2
1
) ( = a c . Indeed, the latter means that one
half of the triples of edges have Alexandrov angles (60,180,180) and the other half have
Alexandrov angles (60,60,180). A simple calculation shows that the average of
∑
=
−
3
1
2
k
k
α π vanishes, hence vanishing scalar curvature.
2.3 Instances of Negatively Curved Graphs
It is well known in Riemannian geometry that a local curvature of uniform sign across
the manifold implies strong global properties. The graph manifestation of this Riemannian
feature is the celebrated Higuchi theorem, saying that, if a cubic graph has positive
combinatorial curvature as per the Higuchi formula uniformly across the graph, it is finite.
Here we reproduce the Riemannian paradigm of checking the local curvature across a
manifold by plotting the histogram of the local combinatorial curvature of the graph.
Should the curvature take value in the negative real axis, it can be asserted that the graph
will manifest some large scale negative curvature properties.
Practically speaking, we cite a few specific situations that naturally lead to such
negatively curved networks:
1. High power transceivers in a wireless sensor network: they have a tendency to
network in a negatively curved graph, while the low power transceivers rather
network in a positively curved graph as shown in Appendix.
2. Delaunay triangulation of nonuniformly distributed vertices: Assume a set of
agents { }
i
v are nonuniformly distributed in
2
R . The Delaunay triangulation of a

18
set { }
i
v of Gauss distributed agents is shown in Figure 2-1, and is negatively
curved as shown in Figure 2-2 by means of the clustering coefficient and in
Figure 2.3 by means of the Alexandrov angles.

Figure 2-1: Delaunay Triangulation of Nonuniformly Distributed Agents

Figure 2-2: Clustering Coefficient Distribution of Delaunay Triangulation

19
-160 -140 -120 -100 -80 -60 -40 -20 0
0
5
10
15
20
25
30
35
40

Figure 2-3: Alexandrov Angles Distribution of Delaunay Triangulation

20
Chapter 3: Large Scale Gromov Curvature
Practically, we are given a set of agents { } V v
i
: = , which attempt to communicate, and
which in doing so define a distance function
+
→ × R V V d : . Typically, the distance
) , (
j i
v v d is the communication delay between
i
v and
j
v , keeping in mind that
i
v and
j
v
might communicate via several “hubs.” As such, ) , ( d V becomes a metric space. ) , ( d V
can also be made a graph by assigning an edge to every pair ) , (
j i
v v of a distance smaller
than a given threshold. The question is to determine the manifold
n
M and its Riemannian
metric g such that the agents can be thought of as operating on the manifold; precisely, the
question is whether there exists a quasi-isometric embedding ) , ( ) , ( g M d V
n
→ . More
specifically, the network can be said to be negatively curved if there exists a quasi-
isometric embedding ) , ( ) , ( g H d V
n
→ in a n-dimenional Riemannian manifold
n
H of
sectional curvature uniformly bounded from above by 0
max
< κ .
The previous definition would still be too strong, since it would enforce the same
distortion bound all across the network, while in some situation the distortion bound would
make sense only in the very large scale.
A class of graphs, referred to as Gromov hyperbolic, behave in the large scale only as
negatively curved Riemannian manifolds. A Gromov hyperbolic graph is a graph that has
the property that its geodesic triangles are “thin,” when looked at a large scale. A geodesic
triangle uvw Δ is a triple of vertices, ) , , ( u w v , along with cost minimizing paths
] , ][ , [ ], , [ u w w v v u . A geodesic triangle is said to be δ-thin if it has an inscribed triangle of a

21
perimeter at most δ . For such a geodesic triangle to “look thin,” it has to be a large scale
triangle, of a diameter much larger than δ . A very large triangle that has an inscribed
triangle of very small diameter looks like a “star,” a particular case of a tree. It is sometimes
intuitively said that a Gromov hyperbolic graph looks like a tree when viewed at a distance.
Unfortunately, no matter how intuitively enlightening this definition is, this definition is
flawed, as not all Gromov hyperbolic graphs are quasi-isometric to trees.
To wrap it altogether, we define a Gromov δ-hyperbolic graph to be a graph such that
uvw Δ ∀ , ] [ ], [ ], [ uv z uw y vw x ∈ ∈ ∈ ∃ such that δ ≤ + + ) , ( ) , ( ) , ( u w d w v d v u d . Obviously,
the latter definition makes sense only for infinite graphs, as all finite graphs enjoy the δ -
hyperbolic property for some finite δ . In the realm of the Internet, which is of awesome
size yet finite, the Gromov hyperbolic property hardly makes any sense, unless we agree on
“how small δ should be.” It can be shown that the application-oriented reformulation of
the Gromov hyperbolic property is , 2 / 3 ) ( diam / < G δ which will henceforth be referred to
as scaled Gromov hyperbolic property. The immediate relevance of the latter concept to
the Internet is the fact that the Growth/preferential attachment process iterated a sufficient
number of times and for some properly chosen connectivity parameter leads to scaled
Gromov hyperbolic graphs.
Geometric optics propagation bouncing on convex obstacles: Assume that in the two-
dimensional Euclidean space
2
R
, we define a set of transceivers
{ }
i
v
along with convex
obstacles. Assume that, at least for some pairs of agents, communication can only be
established by radio wave bouncing on convex obstacles. It can be shown that the

22
embedding is in a negatively curved surface of a genus equal to the number of convex
obstacles.

23
Chapter 4: Mathematical Model and Conjecture
Consider a graph ) , ( E V G = in which every edge e is topologized as the homeomorph
of ] , 0 [ l and endowed with a weight or length ) (e l . A path ) , ( t s p from s to t is a
continuous map G l → ] , 0 [ such that s d s p = ) 0 )( , ( and t l d s p = ) )( , ( . The length of the
path )) , ( ( t s p l is the sum of the lengths of the edges traversed by the path. The distance
) , ( t s d is the infimum of the lengths of all paths joining s to t . A geodesic ] , [ t s is a path
joining s to t of minimum length. A graph is said to be geodesic if V V t s × ∈ ∀ ) , ( there
exists a geodesic ] , [ t s such that ) , ( ]) , ([ t s d t s = l .
The traffic on the graph is driven by a demand measure
+
ℜ → × Λ V V
d
:. The
interpretation of the latter is that, given a source-target pair V V t s × ∈ ) , ( , the demand
) , ( t s
d
Λ is a traffic rate (e.g., number of packets per second) to be transmitted from the
source s to the destination target t. Assume that the routing protocol sends the packets
from the source s to the target t along the path ) , ( t s p with probability )) , ( ( t s p π . As
such, the path ) , ( t s p inherits a traffic rate or intensity measure
)) , ( ( ) , ( )) , ( ( t s p t s t s p
d
π τ Λ =
An edge e laying on the path ) , ( t s p inherits from that path a traffic )) , ( ( t s p τ , and
aggregating this traffic over all source-target pairs and all paths joining the sources to their
destinations yields the traffic rate sustained by the edge e
∑∑
×∈⊇
=
V V tse t s p
t s p e
) ,() , (
)) , ( ( ) ( τ τ

24
Given a connected subgraph G U ⊆ , we define its traffic load to be representative of
the number of packets in it,
∑∑
∈∩ ∈
Λ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= Λ
V t s
d
U t s p e
t
t s p t s e U
,) , (
)) , ( ( ) , ( ) ( ) ( π l
It is also convenient to introduce the subadditive blocking measure, to be
representative of the number of packets transiting through U :
∑∑
∈∈
Λ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= Λ
V t s
d
U t s p e
b
t s p t s e U
,\ ) , (
)) , ( ( ) , ( ) ( ) ( π l
The motivation for the terminology of blocking measure stems form the fact that, in
case of an outage in U, ) (U
b
Λ packets will be blocked from reaching their destinations.
As a metaphor, in the circulatory system, ) (e τ would be the blood flow through the
artery (or vein) e, ) (U
t
Λ would be the amount of blood in the subset U of the vascular
system, while ) (U
b
Λ would be the area affected in case of blood clot or stroke in U .
The load measures are illustrated in the following figure:
5 ) ( = Λ U
t
4 ) ( = Λ U
b
U

Figure 4-1: Assume the Demand Is Uniform, , 1 ) , ( = Λ t s
d
and the Routing Is Optimal, the
Traffic Load in U Is 5 While the Blocking Measure is 4

25
We now define the traffic rate sustained by a node, in such a way as to reveal its deep
connection with betweenness, an archetypical concept of distance geometry, which has
already been claimed to be the central issue in Protein Interaction Network (PIN) [Joy05].
To simplify the notation, we take V V t s t s
d
× ∈ ∀ = Λ ) , ( , 1 ) , ( . Assume the graph is
uniquely geodesic and that the protocol sends the packets along the unique geodesic
[] t s, from the source s to the destination t . Aggregating the demand over all source-target
pairs communicating via the node v yields the traffic rate sustained by v ,
[]{} ) ( : : , # ) , ( ) ( v b t v s t s t s v
t v s
d
= = Λ =
∑
p p
p p τ
In the above, t v s p p means that v is between s and t , meaning that
) , ( ) , ( ) , ( t s d t v d v s d = + , or equivalently, ] , [ t s v ∈ . The rightmost part of the above
equation is called betweenness ) (v b of the vertex v ; it is the number of geodesics passing
through v .
From another point of view, the traffic rate sustained by the node v can be defined as
the sum of the rate of traffic transiting through v , plus the rate of traffic departing from v ,
plus the rate of traffic with destination v . On a graph of order N , the transiting traffic is
) ( 0
2
N , while the traffic with source or target v is ) ( 0 N . It is easily seen on a little thought
that the traffic rate sustained by the node is
) ( 0 ) (
2
1
) (
) (
N e v
v E e
+ =
∑
∈
τ τ
, where ) (v E denotes the set of edges with an end point at v . If we set all edges of
v
E to
be of unit length, the connection with the traffic load measure is easily found as

26
⎟
⎠
⎞
⎜
⎝
⎛
Λ ≈ + Λ = ) (
2
1
) ( 0 ) (
2
1
) (
v t v t
E N E v τ .
Observe that here the traffic is bidirectional and that the paths
G t s p t s p → ))] , ( ( , 0 [ : ) , ( l and G t s p s t p → ))] , ( ( , 0 [ : ) , ( l are not the same, even when
their images are the same. The traffic on an edge e is the traffic aggregated over both
directions. However, since
d
Λ is symmetric, and if π is uniform, ) (e τ is equally
distributed between the traffic in both directions.
Conjecture G
-
: If the graph ) , ( E V G = is negatively curved along with a demand
measure
d
Λ uniformly distributed over V V × , the greedy protocol that sends packets over
optimal routes leads to a very high traffic rate ) (v b over a very small number of vertices v .
Remark: As our benchmark examples will show, this congestion is NOT related to
some heavy-tailed property. Indeed, in most of our examples, the degree of the nodes is
constant, yet one or two nodes carry most of the traffic load. Even more, in such a simple
example as a regular tree, most of the traffic load is carried by the root, which is the lowest
degree node.
Conjecture G
+
: If the graph ) , ( E V G = is nonnegatively curved along with a demand
measure
d
Λ uniformly distributed over V V × , the greedy protocol that sends packets over
optimal routes leads to a nearly uniform traffic rate ) (v b .
If it is anticipated that the network will have extreme congestion points, a natural
problem would be to locate those points. This latter problem leads to our second conjecture,
but before formulating it, we need a few more concepts. The inertia of a graph G with
respect to a vertex v and for uniformly distributed weights is defined as

27
∑
∈
=
V v
i
G
i
v v d v ) , ( ) (
2
φ
Observe that this inertia may be infinite. A center of mass or centroid of the graph G
is defined as a vertex relative to which the inertia is minimum:
) ( min arg ) ( v G cm
G V v
φ
∈
=
To illustrate the above concept, assume that G is the graph of a parallelotope embedded
in a regular lattice of
n
ℜ . The edge length is the one induced by the Euclidean distance and
all vertices have equal mass. The center of mass, as defined above, is ) ( min arg v
G V v
φ
∈
, but
the physical center of mass is ) ( min arg x
G
x
n
φ
ℜ ∈
; observe, however, that the mismatch is no
more than the mesh of the lattice.
We can now formulate our second conjecture saying that the maximum traffic load for
uniformly distributed demand occurs at the center of mass of the network relative to
uniformly distributed weight:
Conjecture G: ) ( ) ( max arg G cm v b = .
Conjecture M
-
: If G is negatively curved, the inertia ) (v
G
φ has a well-defined
minimum and ) ( cm G is unique. (Mathematicians have already proved this result for global
Busemann nonpositively curved spaces.)
Conjecture M
+
: If G is nonnegatively curved, ) ( cm G is not uniquely defined.
In the realm of real data, the inertia ) (v
G
φ is nearly constant with v , and ) ( cm G is not
well-defined.
Clearly, Conjecture
±
M along with Conjecture G would yield Conjecture
±
G .

28
Chapter 5: Benchmark Examples
Several benchmark examples in here will provide support for these conjectures: The
first set of examples is that of planar graphs in which the curvature is dictated by the
valence (degree) of the nodes. We examine the negatively curved cases of valence 7 and 8,
in which significant traffic congestion occurs at the centroid of the graph. Then we contrast
the results with those of a vanishing curvature graph of valence 6, in which the congestion
is more smoothly distributed over all nodes. Towards a more realistic situation, we then
look at a case of mixed valence. We then proceed to positively curved graph of valence <6,
in which the situation is drastically different than in negative curvature, as the congestion is
constant!
It should be noted that congestion might be due to poor implementation of the routing
algorithm, but here we strive to eliminate such potential so as to focus on topology-related
congestion.
5.1 Simulation Results of Negatively Curved Graph
For a graph with degree seven for every node except for the boundary, and if the length
of every edge is equal to one (as shown in Figure 1), and if the routing algorithm is greedy
between any two arbitrary nodes, such as when the routing path is the one with minimum
distance, then the traffic load
) ( 2 ) ( v b E
v t
≈ Λ
on this hyperbolic graph has very high
concentration around a very small subset of the graph close to the center of this hyperbolic
graph.

29

Figure 5-1: The Simple Hyperbolic Graph with Node Degree Seven and Uniform Edge
Length
There are three steps to generate the simple hyperbolic graph with degree seven
everywhere and equal length for all edges.
5.1.1 Simulation Setup: A Simple Hyperbolic Graph Model with Node Degree 7
Everywhere.
There are three steps to generate the simple hyperbolic graph with degree seven
everywhere and equal length for all edges.
Step 1: Initialization: Given an initial/parent node, for example ‘1’, then attach seven
new nodes (‘2-8’) to connect with the parent node ‘1’. Then connect these new nodes to
form a ring like that shown in Figure 5-1.
Step 2: Growth: Attach new nodes to achieve degree seven for their parent’s nodes (i):

30
Given a parent’s node, for example ‘i=2’, then attach new nodes (‘9-12’) to connect with
the parent’s nodes ‘2’ until the node degree of this parent’s node ‘2’ is equal to 7.
Step 3: Connect within the same hierarchy: Connect these new nodes to their neighbors
with the same hierarchy; for example, connect ‘9 to 10,’ ’10 to 11,’ and ’11 to 12’.
Step 4: Finalize the patch: Two more connections are needed to finalize this patch. One
is the connection between the smallest index new node, ‘j=9’ in this case, with the
immediately preceding node ‘j-1=8’; another step is the connection between the biggest
new node, ‘j=12’ in this example, with the node next to the parent’s node ‘i+1=3’.
Recursive step: Recursively cycle through Steps 2 to 4; the graph will be growing up to
a planar hyperbolic graph.
5.1.2 Results: Conjecture G-
By using the Dijkstra algorithm with random pickup
1
, the optimal routing with
minimum length between any two nodes in this graph can be calculated. Figure 5.2 shows
the total number of times a node with index i is visited for all pairs of source-target nodes
in a hyperbolic graph with 100 nodes. The number of times the node i is visited is the
betweenness 2 / ) ( ) (
v t
E i b Λ = of the node i . Remember that those nodes with smaller
indexes represent those nodes closer to the centeroid of the graph. The results shown in
Figure 5-2 indicate that a very small number of vertices carry most of the traffic, as stated

1
Since multiple paths between two nodes in a network may have the same minimum cost, shortest path searching algorithms
such as Dijkstra’s have to be carefully implemented to prevent the artificial congestion resulting from the selection of the
first shortest path every time. While for the negatively curved networks, there are fewer paths with the same minimum cost
than for the positively curved networks, the pitfall of repeatedly selecting the same optimal path is not the major reason for
congestion in a negatively curved network. For positively curved networks on the other hand, the presence of conjugate
points could create an artificial congestion due to poor implementation of Dijkstra’s algorithm. It will therefore be
imperative to consider Dijkstra’s algorithm with random pickup in Section 5.4, when we will study the positive curvature
networks, e.g. Platonic solids. By default, we consider Dijkstra’s algorithm with random pickup, unless otherwise stated.

31
by our Conjecture. Note that this phenomenon transcends the heavy-tailed phenomenon, as
it is not necessary for a node to have high degree for it to carry most of the traffic. In the
above example, all nodes have the same degree; nevertheless, the traffic is concentrated
around one specific node that cannot be singled out by such a local concept as the degree.

0 10 20 30 40 50 60 70 80 90 100
0
500
1000
1500
2000
2500
3000
3500
The vertex index number in the graph
Number of visits
Traffic distribution with random degree7

Figure 5-2: The Traffic Load Distribution of Nodes in a Hyperbolic Network of Valence 7
with 100 Nodes under Random Pickup Routing
5.1.3 Results: Conjecture G
Regarding the second conjecture, the distributions of the traffic (number of visits, top
panel) and the inertia (the average distance squared, bottom panel) for graph of valence 7
are shown in Figure 5-3 It is quite clear from this figure that the maximum congestion
occurs at the point of minimum inertia, which is vertex #1, as shown in Figure 5-1.

32
0 10 20 30 40 50 60 70 80 90 100
0
1000
2000
3000
4000
The vertex number in a graph, the smaller the closer to the center
Number of visits
Traffic Distribution in a Hyperbolic Network (degree 7 ) with Greedy Routing
0 10 20 30 40 50 60 70 80 90 100
0
1000
2000
3000
4000
The vertex number in a hyperbolic networke
The distance square
Distance squre from one node to the other nodes in the graph

Figure 5-3: Traffic and Inertia in the Simple Hyperbolic Graph with Node Degree 7
For negatively curved planar graphs of valence 9, Conjecture I is illustrated in Figure 5-
4, top panel, while Conjecture II is illustrated in the same Figure, bottom panel. Again, it is
very clear that the vertex that has the heaviest congestion (#1) is also the vertex relative to
which the inertia of the graph is minimum. Here, vertex #1 is the start up vertex in the same
recursive construction as the one of Figure 5-1, except that every vertex is given 9 rather
than 7 neighbors.
To draw a fair comparison between the valence 7 and the valence 9 cases, we set the
number of vertices to be the same (100) for both cases. As such, the total demand
∑
× ∈
Λ
V V t s
d
t s
) , (
) , ( is the same for both graphs.

33

0 10 20 30 40 50 60 70 80 90 100
0
2000
4000
6000
The vertex number in a graph, the smaller the closer to the center
Number of visits
Traffic Distribution in a Hyperbolic Network (degree 9) with Greedy Routing
0 10 20 30 40 50 60 70 80 90 100
500
1000
1500
2000
The vertex number in a hyperbolic networke
The distance square
Distance squre from one node to the other nodes in the graph

Figure 5-4: Traffic and Inertia in the Simple Hyperbolic Graph with Node Degree 9
5.1.4 Non-symmetrical Negatively Curved Networks
To further test the conjecture M
-
, especially to eliminate a possible contribution of the
symmetrical structure of the graph to the congestion reported in our previous simulations,
we simulate the traffic and inertia distribution in highly unsymmetrical networks as shown
in Figure 5-5 and Figure 5-6. The position of the heaviest traffic congestion matches the
node with minimum inertia. These results, in further, confirm our conjecture M
-
.

34

Figure 5-5: The Traffic and Inertia Distribution in an Unsymmetrical Network (Vertext#3
Has Maximum Traffic and Minimum Inertia)

35

0 20 40 60 80 100 120
0
1000
2000
3000
4000
X: 14
Y: 3978
The vertex index number
Number of visits
Traffic distribtuion in a non-symmetrical negatively curved networks
X: 4
Y: 3154
0 20 40 60 80 100 120
1000
2000
3000
4000
5000
6000
X: 14
Y: 1165
The vertex index number
The sum of distance square
Distance squre from one node to the other nodes in the graph
X: 3
Y: 1215

Figure 5-6: The Traffic and Inertia Distribution in an Unsymmetrical Network (Vertext#3
Has Maximum Traffic and Minimum Inertia)
5.2 Vanishing Curvature--Planar Graph of Valence 6
From a point, labeled as #1, in the Euclidean plane, we construct a regular triangulation
in which every vertex has valence 6. The triangulation reveals hexagons, which are
“spheres” of radii R r ,..., 1 = with their center at vertex #1; precisely, those spheres are

36
{} r v d v = ) 1 , ( : . As shown in Figure 5-7, the difference of traffic loading among the spheres
is decreased compared with the symmetrical case of valence 7. And we will proof that, as
∞ → R , the traffic loading becomes uniformly distributed.

0 10 20 30 40 50 60 70 80 90 100
200
400
600
800
1000
1200
1400
The vertex index number in the graph
Number of visits
Traffic distribution with random degree 6

Figure 5-7: The Traffic Load Distribution in the Euclidean Plane with Greedy Random
Pickup Routing.
5.3. Nonpositive Curvature
5.3.1 Planar Graph with Random Valence 6, 7 or 8
From vertex #1, construct a planar graph by assigning a random degree (6, 7 or 8) to
each node. The traffic distribution is shown in Figure 5-8a, where the left and right panels
are two independent trials with identical distribution of the valence. The green circles
represent the nodes with degree 6; the red stars represent the nodes with degree 7; and the

37
blue crosses represent the nodes with degree 8. Figure 5-8a demonstrates that, despite the
randomness of the negatively curved nodes and the presence of vanishing curvature nodes,
the traffic distribution is quite heterogeneous with very high traffic load near the center of
mass. The maximum congestion occurs at the point of minimum inertia, which is vertex #1,
as shown in Figure 5-8b. In Figure 5-8c, the maximum traffic occurs at vertex#4, while the
minimum inertia is at vertex #1.

Figure 5-8a: The Traffic Distribution in the Planar Graph with Random Node Degree (6, 7
or 8).
0 10 20 30 40 50 60 70 80 90 100
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
The vertex index number in the graph
Number of visits
Traffic Distribution with random degree 6,7,8
0 10 20 30 40 50 60 70 80 90 100
0
500
1000
1500
2000
2500
3000
The vertex index number in the graph
Number of visits
Traffic distribution with random degree 6, 7, or 8

38

0 10 20 30 40 50 60 70 80 90 100
0
2000
4000
6000
The vertex index
Number of visits
Traffic distribution with random degree 6, 7, or 8 with Greedy Routing
0 10 20 30 40 50 60 70 80 90 100
0
1000
2000
3000
4000
The vertex index
The sum of
distance square
Distance squre from one node to the other nodes in the graph

Figure 5-8b: The Traffic Distribution and the Inertia Distribution (Vertex#1 Has the
Minimum Inertia and Maximum Congestion)

39
0 10 20 30 40 50 60 70 80 90 100
0
1000
2000
3000
The vertex index
Number of visits
Traffic distribution with random degree 6, 7, or 8 with Greedy Routing
X: 4
Y: 2880
0 10 20 30 40 50 60 70 80 90 100
0
1000
2000
3000
The vertex index
The sum of
distance square
Distance squre from one node to the other nodes in the graph
X: 1
Y: 907

Figure 5-8c: The Traffic Distribution and the Inertia Distribution (Vertex#1 Has the
Minimum Inertia and Vertex#4 Has Maximum Congestion)
5.3.2 Comparison I: Between Planar Graphs of Vanishing, Negative and Nonpositive
Curvature
Again, at a root vertex, we attach 8 triangles and from there on we iterate and construct
the various spheres with their center at the root and of various radii r , where r is the
distance ) , ( ⋅ ⋅ d along the edges of the graph. Obviously, these spheres are octagons. Clearly,
for symmetry reasons, the congestion is uniform across the spheres. It turns out that, as a
consequence of negative curvature, the congestion is highest at the root. The results, along
with comparison with the other cases, are shown in Figure 5-9.

40
0 50 100
0
500
1000
1500
Vertex index
Number of visits
Node degree 6
0 50 100
0
1000
2000
3000
4000
Vertex index
Number of visits
Node degree 7
0 50 100
0
1000
2000
3000
4000
Vertex index
Number of visits
Node degree 8
0 50 100
0
1000
2000
3000
Vertex index
Number of visits
Random Node degree 6,7,8

Figure 5-9: The Traffic Distribution in Different Planar Networks, Degree 6 Everywhere,
Degree 7 Everywhere, Degree 8 Everywhere, and Random degree 6/7/8; The Routing
Algorithm Implements Random Pickup
5.3.3 Comparison II: Between Planar Graphs of Vanishing and Negative Curvature
Here we provide an alternative display of the relationship between the inertia and the
traffic. We consider idealized graphs of uniform degree 6, 7, 8, 9, hence of curvature 0,
6
1
3
7 2
2
1
− =
⎟
⎠
⎞
⎜
⎝
⎛
−
π
π
π
,
3
1
− ,
2
1
− , respectively. All graphs have the same size (100). In the
top panel of Figure 5-10, we display the heaviest vertex traffic, that is, ) ( max v b
V v ∈
, versus
the node degree. In the bottom panel of the same figure, we plot the minimum inertia, that
is, ) ( min
V v
v
G
φ
∈
.

41
6 6.5 7 7.5 8 8.5 9
1000
1500
2000
2500
3000
3500
4000
4500
The max traffic as a function of degree
The maximum traffic
6 6.5 7 7.5 8 8.5 9
400
600
800
1000
1200
1400
1600
1800
The minimum inertia as a function of degree
Node degree
The minimum inertia

Figure 5-10: The Maximum Traffic and the Minimum Inertia versus the Node Degree of a
Uniform Graph
6 6.5 7 7.5 8 8.5 9
1000
2000
3000
4000
5000
Traffic in the congestion center
Max traffic
6 6.5 7 7.5 8 8.5 9
4.5
5
5.5
6
6.5
x 10
4
Total traffic in the network
Total traffic
6 6.5 7 7.5 8 8.5 9
500
1000
1500
2000
2500
Total connection links in the network
Node degree
Total edges

Figure 5-11: The Traffic in the Congestion Center, the Total Traffic in the Network and the
Total Connection in the Network
Clearly, as the curvature becomes more and more negative, the vertex carrying the
heaviest traffic in the graph becomes more and more congested, consistently with the

42
inertia at the center of gravity becoming smaller and smaller as shown in the Figure 5-10.
By increasing the node degree, there are more connections in the network, so that the total
traffic in the network decreases, but the traffic is heavier in the congestion center as shown
in the Figure 5-11.
5.4 Positive Curvature: Platonic Solids
To make our case that extreme traffic load at a very limited number of nodes is a
negative curvature phenomenon, we look at the positively curved graphs of the 1-skeleta of
the 5 Platonic solids. More specifically, we show that the node traffic rate ) (v b and the
traffic load ) (
v t
E Λ for uniformly distributed demand measure
d
Λ are uniform. This
sharply contrasts with the traffic distribution in negatively curved graphs.
Recall that the five Platonic solids are the tetrahedron, the cube, the octahedron, the
dodecahedron, and the icosahedron. Among those solids, the tetrahedron, the octahedron
and the icosahedron are the most relevant to our study, because they are made up of
equilateral triangles and the local curvature at a vertex can be defined as
6
1 2
2
1
) (
1
n
v k
n
k
k
− = ⎟
⎠
⎞
⎜
⎝
⎛
− =
∑
=
ω π
π

In the above, n is the number of triangles sharing a common vertex, where we have
taken into consideration the fact that the Alexandroff angle at a vertex of an equilateral
triangle is
6
2 π
ω =
k
. From the above definition, the graphs of the tetrahedron ( 3 = n ), the
octahedron ( 4 = n ), and the icosahedron ( 5 = n ) are all positively curved.

43
The other Platonic solids are made up of regular polygons other than triangles and for
those we have to use the Higuchi-Mohar-Devos formula:
∑
=
+ − =
) deg(
1
1
2
) deg(
1 ) (
v
k k
F
v
v k
, where ) deg(v is the degree of the vertex, ) deg( ,..., 1 : v k F
k
= are the faces at the vertex
vordered so that
1
,
+ k k
F F and
) deg( 1
,
v
F F share an edge, and | |
k
F denotes the number of
vertices of the face. It is easily seen that the latter formula is a generalization of the former.
For the cube, we have 4 , 3 ) deg( = =
k
F v , so that = ) (v k
4
1
>0. For the dodecahedron, we
have 5 , 3 ) deg( = =
k
F v , so that 0
5
2
) ( > = v k .
We will show in the next section that the traffic in those positively curved graphs is
uniform under the shortest-path routing algorithm with random pick-up.
5.5. Dijkstra’s Algorithm with Random Pick-up
We now illustrate Conjecture
+
G by simulating traffic on a positively curved graph.
One difficulty typical of positively curved spaces is that they have conjugate points, i.e.,
pairs of points that can be connected by many geodesics. As such, greedy routing will take
some preferential optimal paths, resulting in nonuniform traffic.
More practically, since there are multiple paths with the same minimum cost between
two nodes in a Platonic solids, the shortest path searching algorithm such as Dijkstra’s has
to be carefully implemented to prevent the artificial congestion resulting from the selection
of the first shortest path every time. As shown in Table 5-1, the normal minimum searching
algorithm, which only picks up the first choice of the shortest path, should be replaced by a

44
randomized pickup mechanism. In further, a multiple random trial has to be implemented
to get the real trend out of the random trial. Figure 5-12 and Figure 5-13 show the
significant difference resulting from the random pickup.

Table 5-1: Dijkstra’s Algorithm with Random Pickup among the Multiple Minimum Paths
Instead of First Minimum Path Pickup

0 2 4 6 8 10 12
0
10
20
30
The vertex index
Number of visits
Traffic Distribtuion in Platonic Solids
0 2 4 6 8 10 12
33
33.5
34
34.5
35
The vertex index
The sum of distance square
Distance squre from one node to the other nodes in the graph

Figure 5-12: Positively Curved Network (Node Degree=5, Icosahedron) without Random
Pickup.

45

0 2 4 6 8 10 12
0
5
10
15
20
The vertex index
Number of visits
Traffic Distribtuion in Platonic Solids
0 2 4 6 8 10 12
33
33.5
34
34.5
35
The vertex index
The sum of distance square
Distance squre from one node to the other nodes in the graph

Figure 5-13: Positively Curved Network (Node Degree=5, Icosahedron) with Random
Pickup among the Multiple Minimum Paths and 10 Trial Average
The shortest-path with random pickup will balance the traffic in positively curved
networks.
For graph with vanishing and negative curvature, we have to invoke the Gauss-Bonnet
theorem on a surface: If 0 ≤ κ everywhere on surface, then two geodesics cannot meet
twice and to bound a disk.
However, for the discrete negatively curved networks, the underlying spaces could still
have multiple geodesics between two vertexes. But the number of such multiple geodesics
will decrease with the curvature while keeping the same weight for every edge. For
example, Figure 5-14 shows two geodesics connecting two vertices. Figure 5-15 shows the
percentage of congestion mitigation as a result of applying the random pickup. Note that
the total traffic is same with and without the random pickup.

46

Figure 5-14: Two Geodesics Connecting a Pair of Vertexes. (Red Circles: the Vertexes
along One Geodesic, Blue Circles: along the Other Geodesic)

Figure 5-15: The Congestion Mitigation Resulting from Randomization of the Pickup

47
5.6 Regular Binary Tree
In this section, we illustrate our conjecture on a binary tree of height n . In case of a
regular tree, the specific vertices, the “traffic core,” is the first layer of the tree, that is, the
layer at a distance of 1 to the root, as we prove in this Note. The latter is yet another
counterexample to the intuitive but erroneous statement that this core is due to the well-
known heavy-tailed phenomenon. In fact, the opposite is true, as the first layer has the same
degree as all of the other layers, except for the root.
In here, we will develop both the simulation and analytical results. The simulation
approach is basically the same as that of the previous example. In the analytical approach,
the overall idea is to first determine the traffic load at the top of the tree; on layers n
and 1 − n . This can easily be done with an intuitive argument. Next, we set up a recursion
of the traffic load as we proceed from the top of the tree to its root.
5.6.1 Basic Definitions
A couple of preliminary notes: The total number of vertices of a binary tree of height n
is 1 2 2 ... 2 2 1
1 2
− = + + + +
+ n n
.
Some nomenclature: By definition, the layer k is the set of vertices v such that
k v v = ) , (
0
word
ρ , where
0
v is the root of the tree. Consider a vertex
k
v on layer k . The
upstream (downstream) traffic is defined to be the traffic that goes to the higher (lower)
layers and is denoted with a subscript + (-). There are various traffic quantities to be
defined:

48
The upstream source traffic
) (k S
+
is the traffic with source at
k
w
and proceeding to
the higher layer k+1. The downstream source traffic
) (k S
−
is the traffic with source at
k
v

and proceeding to the lower layer k-1.
The downstream destination traffic ) (k D
−
is the traffic with destination
k
v coming
from layer k+1. The upstream destination traffic ) (k D
+
is the traffic with destination
k
w
coming from layer k-1.
The upstream straight through traffic ) (k T
+
is the traffic coming from layer k-1,
proceeding to layer k+1 and passing through
k
w unaffected. The downstream straight
through traffic ) (k T
−
is the traffic coming from layer k+1, proceeding to layer k-1 and
passing through
k
v unaffected.
The bouncing traffic ) (k B at layer k is a traffic coming from layer k+1 going through
k
z unaffected and returning to layer k+1.
Observe that
± ± ±
D T S , , B evaluated at the vertex
k
v depend only on the layer on
which the vertex lies, which justifies the notation ) ( ), ( ), ( k D k T k S
± ± ±
,) (k B .
It is easily seen that
2 2 ) ( , 2 2 ) ( ) (
2 2 ) ( , 2 2 ) ( ) (
1 1
1 1
− = − = +
− = − = +
+ −
−
+
+ −
+ −
+
+
+ −
k n n
k n n
k D k D k D
k S k S k S

()
2
1 2 2 ) ( − =
−k n
k B
All of the relevant traffic quantities are shown in Figure 5-12.

49
5.6.2 Traffic Load on Top of Tree (“Initial Conditions”)
Consider a vertex
n
v at the top of the tree, on layer n. This vertex sends traffic at unit
rate to all other vertices and, on the other hand, this vertex receives traffic from all other
vertices. The sending traffic rate of
n
v is
0 ) ( , 2 2 ) (
1
= − =
+
+
−
n S n S
n

and its receiving traffic rate is
0 ) ( , 2 2 ) (
1
= − =
−
+
+
n D n D
n
.
Clearly, there is no bouncing on layer n; hence
. 0 ) ( = n B
There is no straight through traffic either on layer n; hence
0 ) ( =
±
n T
Consider now what is happening between layer n and layer n-1. Let
n n
w v , be two
vertices at layer n such that there exists a vertex
1 − n
v at layer n-1 such that
1 ) , (
1
=
− n n
word
v v ρ and 1 ) , (
1
=
− n n
word
v w ρ . We call bouncing traffic ) 1 ( − n B on
1 − n
v any
traffic from
n
v to
n
w via
1 − n
v . Clearly 2 ) 1 ( = − n B .
5.6.3 Recursion on Traffic
Now that the “initial conditions” ) 1 ( ), 1 ( − − n S n B are set up, we would like to develop
a recursion of the form () )) ( ), ( ( ) 1 ( ), 1 ( k S k B F k S k B = − − .

50
A. Downstream node equation
The conservation of the traffic at
1 − k
v on layer 1 − k yields
() ) 1 ( ) 1 ( ) 1 ( ) ( ) ( 2 − = − − − − +
− − − −
k T k D k B k S k T
In other words
( ) ) ( ) ( 2 ) 1 ( ) 1 ( ) 1 ( k S k T k D k B k T
− − − −
+ = − + − + −
B. Upstream node equation
Conservation of the upstream traffic yields
() ) ( ) 1 ( ) 1 ( ) 1 (
2
1
) ( k D k B k S k T k T
+ + + +
− − + − + − =
In other words,
( ) ) ( ) ( 2 ) 1 ( ) 1 ( ) 1 ( k D k T k B k S k T
+ + + +
+ = − + − + −
C. Downstream/upstream edge equation
We get
() ) ( ) ( ) 1 ( ) 1 ( ) 1 (
2
1
k S k T k B k S k T
− − + +
+ = − + − + −
In other words,
() ) 1 ( ) 1 ( ) 1 ( ) ( ) ( 2 − + − + − = +
+ + − −
k B k S k T k S k T
C. Number of equations/unknowns
B D S , ,
± ±
are known. Hence the only unknowns are
±
T , that is, two unknowns, so that
the three above equations should enough to completely solve the problem. In fact, there is
obviously some redundancy among the three equations. The recursion, linear with forcing
terms, can be set up as

51
) 1 ( ) 1 ( ) ( 2 ) ( 2 ) 1 (
) ( 2 ) 1 ( ) 1 ( ) ( 2 ) 1 (
) 1 ( ) 1 ( ) ( 2 ) ( 2 ) 1 (
− − − − + = −
+ − − − − = −
− − − − + = −
+ − − +
+ + + +
− − − −
k B k S k S k T k T
k D k B k S k T k T
k D k B k S k T k T

D. The total traffic
After the unknowns are calculated, the traffic for one node at layer k is
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( k S k S k D k D k T k T k B k
T + − − + + −
+ + + + + + = Λ
It is shown in Figure 5-16.

Figure 5-16: The Diagram of Traffic among Layers
5.6.4 Results
By using the same Dijkstra algorithm approach as developed in Example I, the traffic
distribution is computed and shown in Figure 5-17. The analytical results obtained by
running the recursion are displayed in Figure 5-18, graphing the traffic at a node versus its
layer. The results displayed in Figure 5-17 and Figure 5-18 are fully consistent. They once

52
again provide support for the conjecture. Another interesting observation is that the highest
traffic congestion is not on the root (the 0th layer, 0 = k ) but on the first layer ( 1 = k ).

0 20 40 60 80 100 120 140
0
2000
4000
6000
8000
10000
12000
X: 2
Y: 1.011e+004
X: 51
Y: 750
X: 25
Y: 1710
X: 15
Y: 3486
X: 5
Y: 6462
X: 1
Y: 8190
The vertex number in a graph, the smaller the closer to the center of the hyperbolic network
Number of visits
Traffic Distribtuion in a Hyperbolic Network with Greedy Routing
X: 100
Y: 252

Figure 5-17: The Traffic Distribution by Using Dijkstra Algorithm and Simulation

53
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
2000
4000
6000
8000
10000
12000
X: 3
Y: 6462
X: 4
Y: 3486
X: 3
Y: 6462
X: 2
Y: 1.011e+004
X: 1
Y: 8190
X: 5
Y: 1710
X: 6
Y: 750

Figure 5-18: The Traffic Load for the Nodes in Different Layer by Analytical Calculation

54
Chapter 6: Mathematical Proofs
6.1 Proof Conjecture G+: Traffic in Positive Curvature
All of these Platonic solids have their symmetry groups Γ. This symmetry group Γ
acts on the vertex set
P
V of the Platonic solid P as a map ) ( ) , ( , v g v g V V
P P
a → × Γ ,
where g is an element of the symmetry group. Recall that the action of a symmetry group
on a space is vertex-transitive if
P
V w v ∈ ∀ , there exists a Γ ∈ g such that ) (v g w = .
It is easy to see that the action of the symmetry groups on the Platonic solids is vertex-
transitive. We prove this as follows: Observe that all Platonic solids except the tetrahedron
have dihedral (rotation) symmetries about axes joining the centers of pairs of opposite faces,
while the tetrahedron has
3
D symmetry about the axis joining a vertex to the center of the
opposite face. Then consider two vertices w v, on a Platonic solid. Join them by a sequence
of consecutive edges. It is easy to see that the vertex at the beginning of an edge can be
moved to the end vertex by a symmetry about the axis perpendicular to the center of a face
comprising the edge.
With the above concept, it is easy to prove that the betweenness is uniform. Let
) (w b
G
denote the betweenness of the node v in the graph G . Then we have
) ( ) ( ) ( ) (
1
v b v b gv b w b
G
G g
G G
= = =
−

The only nontrivial part in the above string of equalities is the second one, where it is
essential that the edge length be uniform. Indeed if ) , ( t s is a pair communicating via gv ,
we have ) , ( ) ), ( ( )) ( , ( t s d t v g d v g s d = + , from which it follows that

55
) , ( ) , ( ) , (
1 1 1 1
t g s g d t g v d v s g d
− − − −
= + , hence there is a pair ) , (
1 1
t g s g
− −
communicating via
v . The proof that the inertia is uniform is essentially the same:
) ( ) ( ) ( ) (
1
v v gv w
G
G g
G G
φ φ φ φ = = =
−

The proof that ) (
v t
E Λ is uniform involves the edge-transitivity of the symmetry group.
Let ) (e
G
τ be the traffic rate on edge e in the graph G . Then
) ( ) ( ) ( ) (
1 1 1 2
1
e e ge e
G
G g
G G
τ τ τ τ = = =
−

Again, in the second inequality, it is essential that the demand be uniform. From the
above, it easily follows that ) ( ) (
w t v t
E E Λ = Λ .
Hence we have the following result: For a uniformly distributed demand measure
+
ℜ → × Λ
P P d
V V : on the squared power of the vertex set of one of the 5 Platonic solids,
the traffic load
+
ℜ →
P
V b: is uniform, for a geodesic routing and provided the traffic is
equally distributed along the many geodesic joining a pair of nodes.
6.2 Proof Conjecture M
-
: Inertia in Hyperbolic Space Model
To justify our numerical results related congestion on planar graphs of uniform valence
of 7, 8, 9, we develop a finite Poincaré disk model. The latter is a faithful model in the
sense that the graphs of valence 7, 8, 9 are quasi-isometric to the Poincaré disk. Recall that
the Poincaré disc {} 1 : < ∈ + = = z C jy x z D inherits its hyperbolic structure through the
metric
()
2
2
2
1
4
z
z dzd
ds
−
= . The latter leads to the area element
()
2
2
1
4
z
dxdy
dA
−
= . In the

56
Poincaré disc, the Laplacian operator is ()
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
−
=
∂
∂
∂
∂
− = Δ
2
2
2
2
2
2
2
2
4
1
1
y x
z
z z
z . A
twice continuously differentiable function f such that 0 = Δf is said to be harmonic. If
0 ≥ Δf , then the function is said to be subharmonic. What motivates the utilization of
(sub)harmonic functions is that they reach their maxima on the boundary of analyticity.
Some results come out a bit more naturally in the conformal upper half plane model
{} 0 : > ∈ + = = y C jy x z U . The metric is
2
2
y
z dzd
ds = , the area element is
2
y
dxdy
dA = ,
and the Laplacian is
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
∂
∂
∂
∂
= Δ
2
2
2
2
2 2
4
y x
y
z z
y .
To avoid some possibly infinite quantities, we will restrict ourselves to the finite
Poincaré disk {} R z C z D
R
< ∈ = : .
We first prove that the origin is the point relative to which the disk has the minimum
inertia and then we show that the maximum congestion occurs at the origin as well.
Minimum disk inertia
Our first concern is the inertia of the finite Poincaré disc relative to the point v ,
∫∫
=
R
D
z dA z v d v ) ( ) , ( ) (
2
φ
More specifically, we want to prove that ) (v φ achieves its minimum at 0 = v . Then we
will show that the congestion for some continuous geometry model is maximum at 0 = v .
This will provide some continuous geometry justification of our conjecture for the graphs
of valence 7, 8, 9.

57
To prove the first assertion, namely, that ) (v φ is subharmonic, we show that
2
) , ( z v d is
subharmonic in v . Indeed, if
2
) , ( z v d is subharmonic, it easily follows that ) (v φ is
subharmonic:
( )
∫∫ ∫∫
≥ Δ = Δ = Δ
r r
D D
z dA z v d z dA z v d v 0 ) ( ) , ( ) ( ) , ( ) (
2 2
φ
By symmetry, ) (v φ is constant on
r
D v ∈ , R r < . By the subharmonic property, ) (v φ
reaches its maximum on
r
D ∂ , R r < ∀ . Hence the minimum is reached at 0 = v .
To prove that
2
) , ( z v d is subharmanic, it suffices to show that ) , ( z v d is subharmonic.
Indeed, assuming that ) , ( z v d is subharmonic in jq p v + = , it easily follows, in either the
Poincaré or the upper half plane model by proper adjustment of λ , that
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
= Δ d d
q
d
p
d
z v d
q p
z v d
2 2
2
2
2
2
2
2
2
2
2
) , (
1
) , (
λ λ

So, if d is subharmonic ( 0 ≥ Δd ) and since 0 ≥ d , it follows that
2
d is subharmonic
(0
2
≥ Δd ).
It remains to show that ) , ( z v d is subharmonic in v . Indeed, the distance in the
Poincaré disk is given by
z v
z v
z v d
−
−
=
1
tanh ) , (
1 -

Clearly,
z v
z v
−
−
1
is holomorphic in v . Hence, as well known,
z v
z v
−
−
1
is sub-harmonic in
v . On the other hand,
-1
tanh is a convex increasing function. The result therefore follows
from a well known fact that the composition of an increasing, convex function and a sub-

58
harmonic function is sub-harmonic.
6.3 Quantitative Measure of Conjecture ± G in Disk
In general, the traffic around a point X of a convex subset of a 2-dimensional metric
space is defined as the following:
()
{}
()
{}
∫∫ ∫∫
∫∫
×
× ∈
∩ =
∩ = Λ
) ( ) (
2
) ( ) ( ) , (
2
' ' ] , [
)) ( ( ) (
1
] , [
)) ( ( ) (
1
) (
R B R B
R B R B t s
t
dydy dxdx t s X length
r B vol R B vol
dsdt t s X length
r B vol R B vol
X

, where ) (R B is a big ball of radius R , and ) (r B is a small ball centered at the point X;
two random points s and t are in the big ball ) (R B ; and the double integral is relative to the
demand measure ) ( ) ( ) , ( t vol s vol t s
d
× = Λ .
The above double integral function can be rewritten as the following:
()
[] {}
() dldl dud Jacobian
t s X length
r B vol R B vol
X
u r uRu r uRr
t
θ
π
×
∩ = Λ
∫∫∫∫
− −−− − −
...
... ,
)) ( ( ) (
1
) (
2 2 222 2 2 2
00
2
00
2

6.3.1 Traffic at the center of a Euclidean Disk, + G :
For the simplest scenario, the convex metric space is a disk, and the point X is the
center.
We first compute the Jacobian related to the change of variables from Cartesian
coordinates to polar coordinates.
Assume the point s and t are at (x, y) and (x’,y’) in Cartesian coordinates. As shown in
Figure 6-1, their corresponding representations in polar coordinates with () ' , , , l l u θ are the
following:

59

Figure 6-1: Traffic in Euclidean Disk
()
() θ θ
π
θ θ
θ θ
π
θ θ
θ θ
π
θ θ
θ θ
π
θ θ
cos ) ' ( sin
2
sin ) ' ( sin '
sin ) ' ( cos
2
cos ) ' ( cos '
cos ) ( sin
2
sin ) ( sin
sin ) ( cos
2
cos ) ( cos
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
u r l u u r l u y
u r l u u r l u x
u r l u u r l u y
u r l u u r l u x
− + − = ⎟
⎠
⎞
⎜
⎝
⎛
− − + + =
− + + = ⎟
⎠
⎞
⎜
⎝
⎛
− − + + =
− + + = ⎟
⎠
⎞
⎜
⎝
⎛
+ − + + =
− + − = ⎟
⎠
⎞
⎜
⎝
⎛
+ − + + =

The Jacobian can be written as follows:

60
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
'
'
'
'
' '
' '
' '
' '
l
y
l
x
l
y
l
x
l
y
l
x
l
y
l
x
y x y x
u
y
u
x
u
y
u
x
Jacobian
θ θ θ θ

( ) ( ) ( ) ( )
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
− + + − + + − − + − − + − −
−
+
−
−
−
−
−
+
=
θ θ
θ θ
θ θ θ θ θ θ θ θ
θ θ θ θ θ θ θ θ
cos sin 0 0
0 0 cos sin
sin ' cos cos ' sin sin cos cos sin
cos sin sin cos cos sin sin cos
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
u r l u u r l u u r l u u r l u
u r
u
u r
u
u r
u
u r
u
'
cos sin 0 0
0 0 cos sin
sin ' cos cos ' sin sin cos cos sin
sin cos sin cos
0 ,
l l
l u l u l u l u
u r
+ =
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
+ + − − − −
≈
→
θ θ
θ θ
θ θ θ θ θ θ θ θ
θ θ θ θ

()
()( )
()( )
( )
R R r
r r
R r
r
u r u r u
R r
du
R r
u r
R
u r r R
r R
du u r l l l l
r R
dldl dud l l u r
X
r
r
R
r
u r r R l l
R
u r uRu r uRr
R t
π π π
π
π
π π
π
π π
θ
π
1
) 1 arcsin(
2
sin
2
4
arcsin
2 2
4
4
lim
2 '
2
1
'
2
1
2
lim
' ' 2
lim ) (
2
1
2
2 2
0
2 2 2
2 2
0
2
2 2
4 2
3
2 2 2 2
2
2
2
0
2 2
'
2 2
2
2
2
00
2
00
2 2
2 2 2 2
2 2 222 2 2 2
= =
⎥
⎦
⎤
⎢
⎣
⎡
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
− − − −
=
− ⎟
⎠
⎞
⎜
⎝
⎛
+
=
+ −
= Λ
−
∞ →
− − − = =
∞ →
− −−− − −
∞ →
∫
∫
∫∫∫∫

61
6.3.2 Conjecture G-: Traffic at the center of a Hyperbolic Disk
As shown in Figure 6-2, the points s and t are at (x, y) and (x’,y’), respectively, in
Cartesian coordinates. It is, however, useful to parameterize the source and target by their
representations in “polar” coordinates ( ) ' , , , r r u θ , where the distances ' , , r r u are hyperbolic.

Figure 6-2: Traffic Model in Hyperbolic Disk
It is important to properly set up the mapping between the coordinates () ' , ' , , y x y x in the
unit disk of the complex plane and the “polar” coordinates ( ) ' , , , r r u θ in the Poincaré disk
model of the Hyperbolic 2-dimensional space of Gauss curvature 1 − = κ . If r r, represent
the Euclidean and hyperbolic measurements, respectively, of the radius, then
⎟
⎠
⎞
⎜
⎝
⎛
= r r
2
1
tanh
.

62
Following the Section 6.2, we have
()
dxdy r
r
dxdy
r
dxdy
dA
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
−
=
2
1
cosh 4
2
1
tanh 1
4
1
4
4
2
2
2
2
,
where the Poincaré disc is { } 1 : < ∈ + = = r C jy x r D .
The Euclidean-hyperbolic coordinate transformation is the following:
()
()
()
()
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − =
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − =
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + =
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + =
'
2
1
tanh ' sin '
'
2
1
tanh ' cos '
2
1
tanh sin
2
1
tanh cos
r y
r x
r y
r x
λ θ
λ θ
θ λ
θ λ

where
r
u
tanh
tanh
cos = λ
and
' tanh
tanh
' cos
r
u
= λ
, per hyperbolic trigonometry in square angle
triangles. Then the Jacobian is

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
'
'
'
'
' '
' '
' '
' '
r
y
r
x
r
y
r
x
r
y
r
x
r
y
r
x
y x y x
u
y
u
x
u
y
u
x
Jacobian
θ θ θ θ
, where
()
()
()() u r
r
u
r
r
u
r
u
x
2
2
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh sin
tanh
tanh
arccos
2
1
tanh sin
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + =
′
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + − =
∂
∂
θ λ
θ λ

63
()
()() u r
r
u
r
u
y
2
2
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh cos ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + − =
∂
∂
θ λ
()
()() u r
r
u
r
u
x
2
2
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' sin
'
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − − =
∂
∂
λ θ
()
()() u r
r
u
r
u
y
2
2
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' cos
'
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − =
∂
∂
λ θ
()
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + − =
∂
∂
r
x
2
1
tanh sin θ λ
θ
, ()
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + =
∂
∂
r
y
2
1
tanh cos θ λ
θ

()
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − − =
∂
∂
'
2
1
tanh ' sin
'
r
x
λ θ
θ
, ()
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − =
∂
∂
'
2
1
tanh ' cos
'
r
y
λ θ
θ

()
()
()()
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + − +
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
′
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+ − +
⎟
⎠
⎞
⎜
⎝
⎛
=
∂
∂
r
r r
u
r
u
r
r
r
u
r
r
x
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
sin ) cos(
2
1
cosh
1
2
1
tanh
tanh
tanh
arccos sin ) cos(
2
1
cosh
1
2 2
2
2
2
θ λ θ λ
θ λ θ λ

() ()
()()
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + + +
⎟
⎠
⎞
⎜
⎝
⎛
=
∂
∂
r
r r
u
r
u
r
r
y
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
2
1
cosh
1
2 2
2
2
θ λ θ λ
0
'
=
∂
∂
r
x
, 0
'
=
∂
∂
r
y

0
'
=
∂
∂
r
x
, 0
'
=
∂
∂
r
y

64
()
()
()()
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ − + −
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
′
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
′
− − − −
⎟
⎠
⎞
⎜
⎝
⎛
=
∂
∂
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
' sin ) ' cos(
'
2
1
cosh
1
'
2
1
tanh
tanh
tanh
arccos ' sin ) ' cos(
'
2
1
cosh
1
'
'
2 2
2
2
2
r
r r
u
r
u
r
r
r
u
r
r
x
λ θ λ θ
λ θ λ θ
() ()
()()
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ ′ − − ′ −
⎟
⎠
⎞
⎜
⎝
⎛
=
∂
∂
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
'
2
1
cosh
1
'
'
2 2
2
2
r
r r
u
r
u
r
r
y
λ θ λ θ
'
' ' '
'
' ' '
'
' ' '
'
' ' '
'
'
'
'
0 0
0 0
' '
' '
r
x
r
x y
u
y y
u
y
r
y
r
x y
u
x x
u
y
r
x
r
y x
u
y y
u
x
r
y
r
y x
u
x x
u
x
r
y
r
x
r
y
r
x
y x y x
u
y
u
x
u
y
u
x
Jacobian
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
− =
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
θ θ θ θ
θ θ θ θ
θ θ θ θ

where,

65
()
()()
()
()
()()
()
() ()
()()
() ()
()()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ ′ − − ′ −
⎟
⎠
⎞
⎜
⎝
⎛
×
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + + +
⎟
⎠
⎞
⎜
⎝
⎛
×
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − +
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ +
− =
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
−
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
'
2
1
cosh
1
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
2
1
cosh
1
2
1
tanh sin
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' sin
'
2
1
tanh ' sin
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh sin
'
' ' '
2 2
2
2
2 2
2
2
2
2
2
2
r
r r
u
r
u
r
r
r r
u
r
u
r
r
u r
r
u
r
r
u r
r
u
r
r
y
r
y x
u
x x
u
x
λ θ λ θ
θ λ θ λ
θ λ λ θ
λ θ θ λ
θ θ

66
()
()()
()
()
()()
()
() ()
()()
()
()()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ − + −
⎟
⎠
⎞
⎜
⎝
⎛
×
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + + +
⎟
⎠
⎞
⎜
⎝
⎛
×
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ −
+
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ +
=
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
' sin ) ' cos(
'
2
1
cosh
1
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
2
1
cosh
1
2
1
tanh sin
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' cos
... '
2
1
tanh ' cos
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh sin
'
' ' '
2 2
2
2
2 2
2
2
2
2
2
2
r
r r
u
r
u
r
r
r r
u
r
u
r
r
u r
r
u
r
r
u r
r
u
r
r
x
r
y x
u
y y
u
x
λ θ λ θ
θ λ θ λ
θ λ λ θ
λ θ θ λ
θ θ

67
()
()()
()
()
()()
()
()
()()
() ()
()()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ ′ − − ′ −
⎟
⎠
⎞
⎜
⎝
⎛
×
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + − +
⎟
⎠
⎞
⎜
⎝
⎛
×
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − +
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ +
=
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
cos sin
'
2
1
cosh
1
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
sin ) cos(
2
1
cosh
1
2
1
tanh cos
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' sin
'
2
1
tanh ' sin
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh cos
'
' ' '
2 2
2
2
2 2
2
2
2
2
2
2
r
r r
u
r
u
r
r
r r
u
r
u
r
r
u r
r
u
r
r
u r
r
u
r
r
y
r
x y
u
x x
u
y
λ θ λ θ
θ λ θ λ
θ λ λ θ
λ θ θ λ
θ θ

68
()
()()
()
()
()()
()
()
()()
()
()()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
′ ′
⋅
⎟
⎠
⎞
⎜
⎝
⎛
′
−
⋅ − + −
⎟
⎠
⎞
⎜
⎝
⎛
×
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅ + − +
⎟
⎠
⎞
⎜
⎝
⎛
×
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
⋅ + ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − +
⎟
⎠
⎞
⎜
⎝
⎛
⋅ − ⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅ +
− =
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
−
'
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
' sin ) ' cos(
'
2
1
cosh
1
2
1
tanh
cosh
1
tanh
tanh
tanh
tanh
1
1
sin ) cos(
2
1
cosh
1
2
1
tanh cos
cosh
1
' tanh
1
' tanh
tanh
1
1
'
2
1
tanh ' cos
'
2
1
tanh ' cos
cosh
1
tanh
1
tanh
tanh
1
1
2
1
tanh cos
'
' ' '
2 2
2
2
2 2
2
2
2
2
2
2
r
r r
u
r
u
r
r
r r
u
r
u
r
r
u r
r
u
r
r
u r
r
u
r
r
x
r
x y
u
y y
u
y
λ θ λ θ
θ λ θ λ
θ λ λ θ
λ θ θ λ
θ θ

69
So that
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂
∂
∂
− =
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
∞ →
'
2
1
cosh
2
1
cosh
1
'
' ' '
'
' ' '
'
' ' '
'
' ' '
'
'
'
'
0 0
0 0
' '
' '
2 2
' ,
r r
O
r
x
r
x y
u
y y
u
y
r
y
r
x y
u
x x
u
y
r
x
r
y x
u
y y
u
x
r
y
r
y x
u
x x
u
x
r
y
r
x
r
y
r
x
y x y x
u
y
u
x
u
y
u
x
Jacobian
r r
θ θ θ θ
θ θ θ θ
θ θ θ θ

()
{}
()
{}()
[] {}()()
∫∫∫ ∫
∫∫ ∫∫
∫∫
× ′ ⋅ × ∩ =
′ ⋅ × ∩ =
∩ = Λ
×
× ∈
R
r
R
r
r
R B R B
R B R B t s
t
r d r d dud Jacobian r r t s X length
dydy dxdx r r t s X length
r B vol R B vol
dsdt t s X length
r B vol R B vol
X
π
θ
2
00
4 4
) ( ) (
4 4
2
) ( ) ( ) , (
2
' cosh cosh 16 ,
' ' cosh cosh 16 ] , [
)) ( ( ) (
1
] , [
)) ( ( ) (
1
) (

70
) (
2
sinh
2
sinh
2
cosh
2
sinh
'
'
2
1
cosh
2
1
cosh
1
...
...
2
sinh 4
2
sinh 4
2
sinh 4
'
2
1
cosh
2
1
cosh 16
cosh
cosh
arccos 2
lim ) (
2 2
2 2
2 2
2 2 2
4 4
C O
R R
R R
O
r d r dud d
r r
O
R r R
r r
u
r
h
X
R
r
R
r
R
r
R
r
R t
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
×
⎟
⎠
⎞
⎜
⎝
⎛
× ⎟
⎠
⎞
⎜
⎝
⎛
× ⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⋅ ⎟
⎠
⎞
⎜
⎝
⎛
× ⎟
⎠
⎞
⎜
⎝
⎛
= Λ
∫∫∫∫
∞ →
θ
π π π

Even after normalization of the traffic load by the square of the area of the big ball
) (R B , the amount of normalized traffic transiting through the small ball ) (r B remains
bounded from below as ∞ → R . This strongly contrasts with the Euclidean case, where the
normalized traffic was going to zero as ∞ → R .

71
Chapter 7: Shortest-path Routing vs. Load Balancing
Routing in Networks of Different Curvature
In the previous chapters, we have shown, from a theoretical point of view, that for
uniformly distributed demand, the greedy routing in negatively curved network causes
congestion over a small number of vertices; moreover, these vertices with maximum traffic
load occur at the center of mass of the network. In this chapter, we more specifically look
at this congestion phenomenon in the practical setting of traffic overload in communication
network. To make this problem more specific and straightforward, we focus our attention
on the multicasting traffic, even though our theorem can be applied to more general
communication network paradigms, such as VoIP and multimedia networking, mobile Ad-
Hoc Networks, wireless sensor network, etc., where the traffic congestion and the routing
algorithm are the big concerns in the design of those communication networks.
Multicasting could involve almost all layers of a communication network. A multicast
task can be performed at the application layer, where a hybrid network is a good model for
this application, as will be presented later. A multicast task can also go systematically
through the physical, links, and network layers, as shown in Figure 7-1. The increasing
popularity of group communication applications such as teleconference and information
dissemination services has led to an increasing interest for the development of multicast
transport protocols. However, these transport protocols could cause congestion collapse if
they are widely used, as they ignore the curvature and are hence prone to the related

72
congestion problems discussed above. The purpose of this chapter is to formulate this
problem and propose a reasonable solution.

Figure 7-1: A Typical Packet Multicasting in Different Layers
Two basic multicast tree algorithms are currently available in the industry: one is the
dense-mode algorithm; the other is the sparse-mode algorithm. Both multicast tree
algorithms are at the heart of the multicast protocols, such as the Distance Vector Multicast
Routing Protocol (DVMRP) in dense-mode, and the Protocol-Independent Multicast
(PIM)-operating in both dense mode and sparse mode. As shown in Figure 7-2a, the dense-
mode uses the source-based tree. It determines a shortest-path tree to all destinations first,
and then uses a reverse shortest-path tree rooted at a source. So the spanning tree starts at
the source and guarantees the lowest cost from a source to all leaves of the trees. The
sparse-mode algorithm uses a shared-tree technique which uses a rendezvous points (RP) to
connect sources and receivers. This rendezvous point acts as the core or root to coordinate

73
forwarding packets from source to receivers under its distribution subtrees, as shown in
Figure 7-2b.

Figure 7-2: Two Methods of Constructing Multicast Algorithms: (a) Dense Mode, Using a
Source-based Tree; (b) Sparse-mode, Using a Shared Tree
We use the Network Simulator (NS2) to build up the traffic congestion environment in
multicast communication. To make our simulation straightforward, we focus our attention
on the congestion vs. network curvature issue by ignoring the dynamic change in the group
membership and using User Datagram Protocol (UDP) as the sources. In further, we apply
the same topology structure (uniform node degree 6, 7 or 8) into the NS2 simulation as the
one already used in Chapter 3. A snapshot of the visualization with NS-2 Nam (the network
Animator) is shown in Figure 7-3. In this figure, the node degree is 6 with a total of 100
nodes in the graph with the node #0 at the center (a similar layout as the one shown in
Figure 5-1).

74

Figure 7-3: A Snapshot of the Visualization of Network with Node Degree 6 by Using the
Nam of NS-2
Other important networking settings are the followings:
Every node in this graph is multicasting to all the other nodes in the network.
The maximum buffer size of the queue in every link between two nodes is 1000 bytes
pks, and every link is a duplex-link with 1Mb bandwidth, with a response time of 2ms.
The size of every file is fixed to 2000 bytes.
The start time of every UDP source is an exponential random variable with average
value 0.01, and the interval time between two successive UDP packets for the source is 2.0
seconds.

75
As shown in Figure 7.4, with the above setting, the network with node degree 8 and
100 nodes has congestion at nodes #0, #2, #3, #4, #5, as revealed by heavy packet drops.
There is no such congestion for the network with node degree 6.
By focusing on the duplex-link between nodes #0 and #1, we have more details about
the congestion, such as the queue size as shown in Figure 7-5, the link utilization as shown
in Figure 7-6, and the queue loss as shown in Figure 7-7. Correspondingly, the queue size,
link utilization, and queue loss for the network with node degree 6 are shown in Figure7-8-
7-10.

Figure 7-4: A Snapshot of the Visualization of Network with Node Degree 8 and Its Packet
Loss by Using NS-2 Nam

76

Figure 7-5: The Queue Size of the Duplex-link between Node #0 and #1 in Node Degree 8
Network

Figure 7-6: The Link Utilization of the Duplex-link between Node #0 and #1 in Node
Degree 8 Network

77

Figure 7-7: The Packets Loss of the Duplex-link Between Nodes #0 and #1 in Node Degree
8 Network

Figure 7-8: The Queue Size of the Duplex-link Between Node #0 and #1 in Node Degree 6
Network

78

Figure 7-9: The Link Utilization of the Duplex-link between Node #0 and #1 in Node
Degree 6 Network

Figure 7-10: The Packets Loss of the Duplex-link between Node #0 and #1 in Node Degree
6 Network

79
Load Balancing Routing
Load balancing algorithms are widely used to curb the congestion. For example, Cisco
IOS ® router software has built-in load balancing functionality, and is available across all
router platforms. It allows a router to use multiple equal cost paths to a destination when
forwarding packets. The fundamental mechanism is as follows: When the router must
select a route from many with the same administrative distance, the routers choose the path
with the lowest congestion cost to the destination. In further, one can select load-balancing
to work per-destination or per-packet. As shown in Table 7-1 [Cisco05], the position of the
asterisk (*) points to the interface over which the next packet/destination-based flow is
send; and the asterisk (*) keeps rotating among the equal cost paths each time a packet/flow
is served.

Table 7-1: In Load Balancing, the Asterisk (*) Keeps Rotating among the Equal Cast Paths
Table 7-1 is very similar to Table 5-1 and can achieve load balancing in the positively
curved networks as shown in Figure 5-13. However, in most cases, the negatively curved
networks have worst congestion problem, and the current load-balancing algorithm can not
alleviate it. The reason is this: first of all, there are no such multiple paths with same
administrative distance, since negatively curved networks have no conjugate points as the

80
positively curved networks have. Second, even if we allow for the quasi-optimal paths,
there are still too close to bypass the congestion points.
So, in here, we propose a curvature based load-balancing algorithm. The system
diagram is shown in Figure 7-11. The curvature κ of the network is used to control a
switch. If the network is nonnegatively curved, 0 ≥ κ , the weight of the edges in the
network is the administrative distance; and the shortest path is calculated based on that.
Therefore, traditional load-balancing is used as we mentioned above. If the network is
negatively curved, the weight of the edge between two directly connected vertex i # and
vertex j # is reassigned to be:
()
⎪
⎩
⎪
⎨
⎧
∞
= ∗ • ∗ =
=
−
∑ ∑
otherwise
j i d if j d i d
j i weight
1 ) , ( ) , ( ) , (
) , (
1
2 2

where d is the administrative distance.
The virtual graph is generated with these weighted edges instead of the administrative
distances. The inertia distribution will be flat and the curvature will tend to be positive in
this virtual graph since the edges close to minimum inertia vertex (with heaviest traffic) of
original graph are assigned larger weight to curb the routing. (The connection with the
celebrated Yamabe flow problem is obvious.)

81

Figure 7-11: The System Diagram of Curvature Based Load-balancing
Figure 7-12 compares the traffic distribution with and without the curvature based load-
balancing. In this experiment, we use the node degree 7 network. The heaviest traffic drops
from 3340 to 1756 after the curvature based load-balancing. It is a 47% decrease. Since the
paths have to be detoured from the congestion vertex through extra routers, it will cause an
increase of the total traffic in the network. The total traffic with the load-balancing is 69126
compared to 53964 without the load-balancing. It is a 28% increase. Figure 7-13 compares
the typical routings with and without the load-balancing.
Considering the price paid in total traffic for applying load-balancing to avoid
congestion, a method to avoid this drawback consists in connecting the routers to form a
nonnegatively curved network at the beginning. An idea for generating positively or
negatively curved networks from the same building block is discussed in Appendix section
2.2. A more practical method to generate a nonnegatively curved network is considering an
heterogeneous hybrid network to avoid the global negative curvature properties.

82

Figure 7-12: Traffic Distribution of Node Degree 7 Network (Left: without Load-balancing;
Right: with Load-balancing)

Figure 7-13: Routing with and without the Curvature Based Load Balancing (Red Circle
without Load Balancing, Blue Circle with Load Balancing)

83
Chapter 8: Summary and Future Works
We have proposed to use the concept of curvature of coarse geometry as the key to
analyze the traffic pattern in networks, especially in negatively curved networks. We have
found that the Alexandrov angles, the clustering coefficient, and the Gromov
δ
are
consistently providing the consistent curvature information. The importance of the latter is
that it provides the quintessence of the topological structure of a network. With that
curvature metric, networks can be classified into three fundamental structures: negatively
curved networks, positively curved networks, and hybrid networks. Networks with
different curvature have different behaviors as far as traffic, random walk, percolation
processes are concerned. New control paradigms can be envisioned when it comes to
changing the curvature of networks with minimal cost.

84
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[Wikipedia08] Definition of the Congestion. Wikipedia, 2008

87
Appendix: Applications
The thesis of the Appendix is to present the applications of traffic patterns and
networking curvature. The concept of curvature and the corresponding traffic pattern have
profound ramifications in a variety of problems. We consider several applications that are
somewhat outside beaten tracks—the piracy propagation problem, which is already well
developed and has already been published, the parallel computation problem and its
application in VLSI/FPGA connection problem, sensor networking problem and
waterflooding in the petroleum engineering which are not in as mature a stage as the traffic
congestion in the communication, but which nevertheless appears promising in the way it
puts to use the curvature concept and traffic control.
1 Tracking and Mitigating Piracy
2

The American motion picture industry and software companies have been struggling
with the problem of the illegal copying and distribution of its products across the world,
particularly in China. The piracy problem in China has become a hurdle to the American
companies who want to enter into the most important potential market in the world.
Although U.S government and enterprise have called for Chinese government to control
the piracy propagation more effectively, they feel that more reliable data should be
obtained to evaluate the damage before entering into a confrontation with the Chinese
government. However, Chinese piracy is a multi-facet problem associated with its
economic and social infrastructures. At the same time, obtaining the plausible statistics that

2
This project is supported by USC Chinese Institute.

88
can foresee the extent of these socio-economic problems is extraordinarily difficult. Up to a
certain extent, this issue has affected the predictability of the U.S.-China relationship. This
project proposes to use the Chinese market as a benchmark case study of the piracy; it is
specifically aimed at recovering statistics data from limited source and noisy measurement,
and at providing several efficient anti-piracy strategies.
The piracy propagation issue has been studied in our previous research [Lou06]. The
new mathematical concept developed in [Lou06] was that of hybrid network, that is, a
network that has a Small-world with positive curvature (Social Network) and a Scale-free
with negative curvature (Internet) components connected by a switching mechanism, as
shown in Figure A-1 and Figure 1-4.
The result in [Lou06] showed that the propagation model on this hybrid network
matches real piracy propagation traces of movies that were in the U.S. box office top 50
between January 1, 2002 and June 27, 2003 [Byer03], as shown in Figure A-2.

Figure A-1: Piracy Propagation Model in the Decentralized Small-world Network (Left)
and Centralized Scale-free Network (Right)

89

Figure A-2: Left, Distribution of Theater/Internet Release Time Lags for Samples in AT&T
Data Set. Week 0 is the Week a Movie was First Released in U.S. Theaters. Right:
Simulation Results of a Piracy Index Increment Distribution
Much more challenging, however, are the asymptotic problems in the limit of very
large graphs and very long run times. In the abstracted setting of infinite graphs, a random
walk piracy propagation is either transient (the propagation dies by itself, as most Internet
worms thus far) or escapes to infinity (the propagation is “explosive,” something no yet
observed in the Internet, but frightening at the thought it could happen) depending on
whether or not the effective resistance between the piracy seed and the boundary at infinity
of the graph [Jonckheere07d] is infinite or finite. The latter problems have been solved in
the literature, for such homogeneous graphs as trees, regular lattices, etc. Here the
challenge is to extent those results to hybrid graphs.
Using the Cheeger isoperimetric constant bi-partition methods introduced in later, this
hybrid graphs (joint graph with Scale Free and Small World graph) will be isolated as
shown in Figure A-3. As shown in Figure A-3, even for a very small potion of the hybrid
graphs, (the graph order is around 60), the cheeger isoperimetric constant still can be used
to bi-partition. However, the degree measurement is not properate in the bi-partion since
the average degree in the Scale Free graphs is 9.78, and it is 10 in the Small World graphs.

90

Figure A-3: The Joint Graph with Scale Free and Small World Graphs Is Isolated by
Cheeger Isoperimetric Bi-partition. (Red nodes Are Belong to the Small World Graph; and
the Blue Nodes Are the Vertexes of the Scale Free Graph)
In summary, Internet has connected the world together. However, the traditional social
networks and legislative regulations with different structures and rules than the Internet still
exist. The hybrid network paradigm will be an innovative research approach for the
mathematician, the engineer, and the public policy maker in the immediate future,
especially, after people discovered its inability to interpret signals of a market and social
impact of Internet, such as Google, Facebook, Youtube and Myspaces, that behaviors
differently and is governed by different rules. This project will use the piracy problem as a
benchmark case study, to discover the predictability and controllability behind the
complicated relationship between the social network and Internet.

91
2 Optimize Parallel Computing Performance
Traditionally, software has been written as serial processing and executed by a single
Central Processing Unit. Because the von Neuman architecture is centralized, it can be
thought of as of negative curvature. The new parallel computing platforms, such as
networked parallel computer “grids,” the UltraScience Net of the DOE, and multi-
processor computers, can overcome the bottle-neck of a single computer and substantially
save on the computation time by changing the curvature to positive. The same concept can
be also applied to the power grid problem. Using the results of this study, we are cleared
that the decentralization and parallelism are not the fundamental solution to avoid the
congestion. The fundamental way to boost the parallel computing is adopting a positively
curved architecture as opposed to the traditional negatively curved one.
In detail, a typical networked parallel computer “grid” is shown in Figure A-4. The
configuration and computational resource of each unit in this networking may be different.
An example of a parallel “hello world” program with the instruction flow is shown in
Figure A-5. The parallelized computation tasks are distributed to every unit. If all tasks are
subject to a barrier synchronization point, the slowest task will determine the overall
performance. Figure A-6 shows a 802.11 receiver module with a Viterbi decoder in Xilinx
FPGA. For designing high-end FPGAs, the clusters of these logic blocks need some
optimal partitioning and routing algorithms to maximize the performance and at the same
time to satisfy the hardware restrictions. The researchers in the industry have developed
many efficient placement and global routing algorithms for the hierarchical FPGA
(HFPGAs) [Tang00]. Normally, the so-called ‘min-cut’ techniques are used to recursively
partition a cluster into sub-clusters to facilitate the architectural design of the HPGAs.

92
However, these algorithms have some hitherto hidden interpretation in terms of curvature
and Cheeger Isoperimetric concepts.
Mathematically, the optimization objective is to maximize the curvature/Cheeger
isoperimetric constant/effective resistance of the network of parallel computation resources
and avoid the traffic congestion in the computation flow. This optimization algorithm can
be realized in the commercial and industrially popular System-on-a-chip (SOC)
application by means of Field-Programmable Gate Array (FPGA) and the Hardware
Description Languages (HDL). With the reconfigurable computing and parallel computing
architecture, a SOC will make optimized adaptive changes to the data path and the control
flow corresponding to different types of computational demands.

Figure A-4: A Typical Networked Parallel Computer “Grid”

Figure A-5: An Example of Parallel “Hello World” Program with the Instruction Flow,
from: http://www.llnl.gov/computing/tutorials/parallel_comp/

93

Figure A-6a: Hierarchy View of a 802.11 Receiver Module with a Viterbi Decoder in
Xilinx FPGA

Figure A-6b: Initial Placement of a 802.11 Receiver Module with a Viterbi Decoder in
Xilinx FPGA

94
2.1 A Case for Positive Curvature
A generic graph partitioning algorithm has been proposed to minimize the number of
wires connecting the various boards/sub-networks of VLSI/FPGA. The proposed
algorithms have been based on (well-thought) heuristics. Here we propose a more
mathematically oriented approach based on the Cheeger isoperimetric constant. In a certain
sense, this mathematical concept embodies the statement made in [Tang00, Section 3] that
“the balance between the edge cuts and the size of the partition should be considered.”
Somehow, the number of wires between two subsets of a bi-partitioning is related to the
Cheeger isoperimetric constant of the graph, ) (G h . From there on, a recursive bi-
partitioning is implemented, from which it follows that the grand total number of wires is
the sum of the Cheeger isoperimetric constants of a nested sequences of subsets of the
graph. In a traditional sense, the large ) (G h the more negatively curved the graph is. (This
concept is well known and documented for infinite graphs, but needs some adjustment
when it applies to finite graphs.) Traditionally, FPGA’s have been designed with a
centralized, von Neumann like architecture. This centralized design has traditionally made
FPGA negatively curved. However, since here our aim is to minimize the Cheeger
isoperimetric constant, it appears that a positively curved architecture is better.
Consider a finite graph ) , ( E V G = . The Cheeger isoperimetric constant ) (G h of the
graph is defined as
| |
inf ) (
2 / | | | |
S
S
G h
V S
V S
∂
=
≤
⊂

, where the boundary S ∂ of a subset of vertices is defined as the set of edges from S to its

95
complement, that is, S . From a slightly different standpoint, the Cheeger constant refers to
a bi-partition S S V ∪ = .
For an infinite graph, the restriction on the cardinality of S is removed. A similar
definition holds for an r-dimensional manifold:
) ( vol
) ( vol
inf ) (
r
1 - r
K
K
M h
M K
r
∂
=
⊆

where the infimum is extended over all compact subsets K with rectifiable boundary.
If a space has 0 > h , it has a linear isoperimetric inequality and is hence hyperbolic. To
be slightly more explicit,
) ( vol
) (
1
) ( vol
1 - r r
K
M h
K
r
∂ ≤
Conversely, a Gromov hyperbolic manifold or graph need not have . 0 > h Positive
Cheeger constant of a Gromov hyperbolic space requires, among other things, some
condition at infinity. If 0 > h can be secured, then the manifold or graph has exponential
growth. The converse does not however hold; some Lie groups have 0 = h yet exponential
growth.
To decide whether a finite graph is negatively or positively curved from the Cheeger
isoperimetric constant is a problem that has not yet been completely resolved. Probably an
approach along the lines of the scaled Gromov hyperbolic δ should be pursued. Here we
will limit ourselves to the intuitive statement that the lower the Cheeger constant the more
the graph is towards positive curvature.
From the application-oriented VLSI/FPGA point of view, the criterion is the number of
wires, mathematically embodied in the concept of boundary. Hence the relevant way to

96
rewrite the Cheeger isoperimetric inequality is as
2 / , , ) ( V S V S V G h S S ≤ ⊆ ∀ ≥ ∂ = ∂
It follows that the number of wires ( V G h S S ) ( ≥ ∂ = ∂ ) is always bounded from
below by | | ) ( V G h . Hence it makes sense to reach the equality by doing a Cheeger
partitioning.
The idea is to recursively partition the graph so as to reach the Cheeger isoperimetric
constant at every step. The basic recursion is to reach the Cheeger isoperimetric
partitioning of the large subset of the previous partitioning.
Partition ) , ( E V G = as
0 0
L S V ∪ = , where
0
S , the large subset, is defined as
2
| |
| |
0
V
S ≤ and | |
) (
1
| |
0 0
S
G h
S ∂ = . (For recursive purpose, it might be useful to set
V L =
−1
.)
The basic recursive step is the re-partitioning
1 1 + +
∪ =
i i i
S L L , where
2
| |
| |
1
i
i
L
S ≤
+
and
| |
)) ( (
1
| |
1 1 + +
∂ =
i
i
i
S
L G h
S , where ) (
i
L G denotes the sub-graph of G made up with the
vertex set
i
L . (For recursive purpose, it might be useful to set G L G =
−
) (
1
.)
It is easily seen that the algorithm terminates when 1 =
N
S .
The culprit is that the grand total number of wires is

1
1
1
)) ( (
+
−
− =
∑ i
N
i
i
S L G h

97
At every step, the number of wired balanced relative to the size of the partitions
[Tang00] is minimized, but of course we do not know as yet whether the overall
partitioning is optimal, and if not how far (how close?) we are to overall optimality. With
the discussion with our colleagues in the Bell lab and USC, the suggestions are focused on
using eigenvalue spectra for graph.
2.2 Partition and Combination
From the above, we have demonstrated that the Cheeger isoperimetric constant can be
used to partition a graph into sub-graphs with different curvature characteristics, as shown
in Figure A-3 and other results in this section. As we recursively partition the graph, the
curvature characteristics of these refined sub-graphs will be merged into a common one, as
shown in Figure A-7. Figure A-8a and Figure A-8b show the different partition patterns in
negatively curved sub-graph (upper half plane) and positively curved sub-graph (under half
plane). So another reverse application is how to combine these common building blocks
together to generate different curvature graphs. In here, we will study the numerical
partition problem.

98
1 2 3 4 5 6 7 8 9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Cheeger Constant in
Small-world graph bi-partition
Cheeger Constant in
Scale-free graph bi-partition

Figure A-7: Cheeger Constant Evolution of the Sub-graphs in Recursive Bi-partitioning
(the Order of the Original Graph is 1000)
0 5 10 15 20 25 30 35 40 45
0
5
10
15
20
25
30
35
40
45

Figure A-8a: (by Hilbert Cube Method) 1
st
Bi-Partition, Cut the Hybrid Piracy Network
into 2 Parts; the Cut is Consistent with the Decomposition of the Network in the Scale-free
(Blue) and the Small-world (Green) Subnetworks

99
0 5 10 15 20 25 30 35 40 45
0
5
10
15
20
25
30
35
40
45

Figure A-8b: (by Hilbert Cube Method) 2
nd
Bi-Partition, Do a Cheeger Partitioning of the
Two Sub-networks Resulting from the First Cheeger bipartitioning; This Leads to a
Decomposition into 4 Parts: the Previous Scale-free Part is Decomposed into Blue and
Green Sub-networks and the Previous Small-world part Is Decomposed into a Red and a
Cyanic Blue Sub-networks

2.2.1 Heuristic Method
Reference [Grady06] shows how to minimize the Cheeger isoperimetric constant
r x
Lx x
h
T
T
x G
min = of the graph by a heuristic, but well thought of, method; but here we use a
different argument to reach the Cheeger isoperimetric constant.
First, from the electrical interpretation of L, it is singular, as 0 = Lr . To avoid the
singularity of the matrix L, we select one reference node with maximum node degree,
remove the row and column of L corresponding to this node to get the matrix
0
L . By the
same token, this yields the vectors
0
x and
0
r . However, to simplify the notation, we will

100
drop the subscript 0. With this convention, L is nonsingular and, from its passive network
interpretation, 0 > =
T
L L . It is clear that relaxing the indicator property of x would yield
the absurd solution 0 ↓ x , which is not acceptable since it would mean a trivial partitioning
like ∅ ∪ = V V .
We could proceed heuristically as follows: Set r Lx μ = , in which case μ =
G
h . The
limiting factor as to how low μ =
G
h can be dropped is whether the corresponding solution
r L x
1 −
= μ has an indicator interpretation. At this stage, we follow [Ghosh06]: we find the
median value of {}
0
\ : ) ( v V i i x ∈ , x , and then declare S i ∈ whenever x i x ≥ ) (. The same
method yields quite similar results when we use d instead of r .
The electrical interpretation of
0
L is
0
1
0 0
I L V
−
= , where
0
V is the vector of node
potentials relative to the ground potential of the reference node and
0
I is the vector of
external currents injected at the nodes, with no current injected at the reference node.
0
1
0 0
r L x
−
= is referred to as the potential for a graph with weight from node i to j equals
to
ij
w ., and
0
r is corresponding to the current input into this graph. In this interpretation, the
Cheeger constant is equivalent to find the bottle neck of the graph which causes large
effective resistance.
Intuitively, the optimal results obtained in paper [Ghosh06] are increasing the
conductance of these bottle necks by weight relocation.

101
2.2.2 Hilbert Cube Method
Searching over all indicator functions { } 1 , 0 \ :
0
→ v V x might be computationally
prohibitive, so that it is desirable to develop a continuum approach. Here we develop one
such approach, the distinguishing feature of which is that it is rigorous, as opposed to the
heuristics of the previous one.
Clearly, the condition that x is an indicator function means that ) (i x is restricted to be
a vertex of the cube
1 | |
] 1 , 0 [
− V
. For an infinite graph, the latter would be the famous Hilbert
cube. The key idea here is that the cube
1 | |
] 1 , 0 [
− V
can be approximated by a ball
⎟
⎠
⎞
⎜
⎝
⎛
−
2
4
1 | |
r
B
V

with its center at
2
r
and radius
4
1 | | − V
. As such, the boundary of the cube can be
approximated by the boundary of the ball, and the vertices of the cube can be approximated
by the sphere
⎭
⎬
⎫
⎩
⎨
⎧
−
= − ℜ ∈
−
2
1 | |
2
:
1 | |
V r
x x
V
. It should be clear by now that the Cheeger
isoperimetric partitioning can be viewed as a constrained optimization problem, to be
solved by canceling the gradient of an augmented functional
0
2 2
2 1
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
− + ∇ r x
r
x
r
x
r x
Lx x
T
T
T
T
x
λ λ
, where
1
λ and
2
λ are the Lagrange multiplier, and the last equality condition is that
2 / V r x
T
= . Canceling the gradient yields
0
2
2
) (
) ( ) 2 (
2 1
2
= +
⎟
⎠
⎞
⎜
⎝
⎛
− +
−
r
r
x
r x
Lx x r r x Lx
T
T T
λ λ

102
In other words,
() ()
0
) (
2
2 ) ( ) 2 (
2
2
2
2
1
=
+
⎟
⎠
⎞
⎜
⎝
⎛
− + −
r x
r x r r x
r
x Lx x r r x Lx
T
T T T T
λ λ

Since r x
T
, Lx x
T
are scalars, it follows that Lx , r , and x must be linearly dependent:
x r Lx ν μ + = . It is indeed readily seen that the coefficient of Lx could not vanish for
otherwise the partitioning would be the trivial one. Once ν μ, are determined, the solution
is found as () r I L x μ ν
1 −
− = and the final step would be to find the nearest vertex.
Unfortunately, ν μ, cannot be determined independently of
1
λ and
2
λ , so that we need to
find four relationships among ν μ λ λ , , ,
2 1
.
The first such relation comes from
4
1 | |
2
2
−
= −
V r
x , which yields
()
4
1 | |
2
2
1
−
= − −
−
V r
r I L μ ν
The second equation is 2 / V r x
T
= .
The next relationship is obtained by premultiplying the numerator of the gradient of the
augmented functional by
T
x , which yields
() ( ) ( ) ( ) ( ) ( ) ( ) 0 2
1
2
1
1
1 1
= − + − − + − −
− − − −
r I L r r r r I L r I L L I L r
T T
ν μ λ μ ν λ μ ν ν μ
A fourth relation is obtained by premultiplication of the numerator of the gradient of
the augmented functional by
2
T
T
r
x − , which yields

103
() ( ) ()
() ( ) ()
() () () () () 0
2 4
1 | |
2
1 | |
2
1
2
1
2
1
1
1 1
1 1 1
= −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− − + −
−
+
− − −
⎟
⎠
⎞
⎜
⎝
⎛ −
+ − − −
− − −
− −
− − −
r I L r r
r
I L r r I L r
V
r I L r r I L L r
V
r I L r r I L L I L r
T
T
T T
T T
T T
ν μ λ ν μ ν μ λ
ν μ μ ν
ν μ μ ν ν μ

The above four relations allow for the solution in ν μ λ λ , , ,
2 1
and hence for the overall
solution x . The binary result of x is generated by filtering the value of x with the medium
value of x .
2.2.3 Comparing the Partitioning Results: Heuristic Method vs. Hilbert Cube Method
The partitioning results by using the Heuristic method are as same as by the Hilbert
cube method. However, the Hilbert cube method with strict theoretical deduction is more
favored by mathematicians.
2.3 Some Further Speculations
It would be interesting to relate the above concept to the Cray-1 computer architecture.
The processors of the Cray-1 computer are distributed on a cylinder, the basis of which is
horseshoe or C-shaped. Speed-dependent portions of the system are placed on the “inside
face” of the cylinder where the wire-lengths are shorter. It would be interesting to compute
the curvature of this interconnection. It is fair to conjecture that it is positive. The Cray-1
architecture indeed resembles the Small World and since the latter is positively curved, so
is the Cray-1 expected to be.
The Cray T3D (“Torus 3-Dimensional”), on the other hand, is connected to form a
three-dimensional torus network topology and, again, it would be interesting to compute its

104
curvature and attempt to relate it to its performance. From a strict mathematical viewpoint,
the curvature of the 3-torus
3 3 3
/ Z T ℜ = vanishes.
Similar considerations can be made for the brain. The gray matter consists of neurons
distributed peripherally and making the cerebral cortex. The white matter on the other hand
is the extremely dense network of myelinated nerve fibers interconnecting the various
neurons. Again, the fundamental architecture of processors distributed peripherally and
shortcut links following diametrical paths is there.
We could construct the following mathematical model: Consider a sphere
2
S with
vertices arranged in a Kepler tessellation pattern. This graph acquires a metric closely
related to the metric induced on the sphere by the usual Euclidean metric. Now, let us
connect diametrically opposed vertices with links of length small compared with the typical
length of the edges connecting neighboring vertices on the sphere. At the limit, if the length
of the diameter links is vanishing, we would obtain the real projective plane,
2
P ℜ which is
positively curved, because its universal cover is the 2-sphere
2
S . Invoking a general
continuity argument, it seems reasonable to conjecture that the curvature will still be
positive for diameter links of a cost small enough.
It is also interesting to observe that the spinal cord has the opposite architecture: the
nerve fibers are at the periphery whereas the neurons are centrally located.
3 Localization and Synchronization in Sensor Network
In our works [Lou07], we applied the curvature concepts (Alexandrov angles and
clustering coefficient) to sensor networks. Three cases of sensor networks were analyzed.
Probably the most consistent pattern that has been observed is that retaining links of better

105
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.4
0.45
0.5
0.55
0.6
0.65
The value of threshold
The Average of Clutering Coefficient
The Average of Clusting Coefficient as a Function of Threshold
Combination of
Type I and Type II Nodes
Type I Nodes Only
Type II Nodes Only
and better quality makes the network transits from positive curvature (“meshed” network)
to negative curvature (“core concentric” network), as shown in Figure A-9.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-10
-8
-6
-4
-2
0
2
x 10
4
The Average of the Alexandrov curvature as Function of Threshold.
(Weight=1-0.99PRR, Zero area=10
-6
)
Threshold
The Average of Alexndrov curvature
Data Set 06B
Data Set 06A

Figure A-9: Right, the Average of the Alexandrov Curvature as Function of Threshold;
Left, Average of Clustering Coefficient as a Function of Threshold for Data Set 06A
Among the features already revealed by the curvature, one will retain the difference
between the Type I and Type II sensors, the effect of the better antenna design in the Type
II sensor, and finally the tendency of the network to go from meshed to core concentric as
links of better and better quality are retained. On a more general tone, since the
fundamental Riemannian structure had been reproduced on a graph, such Riemannian
features as the stability of the geodesics in negative curvature would map to a resiliency of
the communications routes under changes in the propagation parameters, typical of extreme
environments.
One observation made on the sensor network examples is that the distribution of
clustering coefficients has its mean on the negative curvature side, but has a power law
distribution on the positive curvature side. In simple terms, the sensor network has an
overall negative curvature, with some points of positive curvature.

106
In this sensor network, we applied the traffic analysis (uniformly distributed demand,
greedy routing and the Dijkstra’s algorithm with random pickup). As shown in Figure A-10,
the sensor network with threshold 0.5 has heaviest congestion (vertex #88 and #81) than
the congestion in the network with threshold 0.1. (vertex #91 and #86), and the curvature of
the network with threshold 0.5 is more negative than with threshold 0.1. As shown in
Figure A-11, the most convincing result is that by the vertex (#88) with heaviest congestion
has the minimum inertia in the more negatively curved graph with threshold 0.5; and the
minimum inertia and traffic congestion concepts are becoming ambiguous once the
curvature turns to be positive, such as the network with threshold 0.1.
Unfortunately, the vertex#88 in the network with threshold 0.5 does have maximum
node degree 58 (mean node degree is 27.82 and the minimum is 1). However, as we
proofed in Chapter 6, the traffic congestion is not because of the heavy-tail phenomena or
node degree. The reason of this result and other peoples’ previous confusion are the
interlaced mixing of the vertexes with positively and negatively curvature. It’s another
example of hybrid networks which we proposed before. Since this mixture, the negatively
curved network is cut to small pieces by a few of nodes with positively curvature, and the
local measurement, such as node degree, is feint related to the congestion.

107
0 10 20 30 40 50 60 70 80 90 100
100
200
300
400
500
600
700
X: 88
Y: 656
X: 81
Y: 578
X: 91
Y: 468
X: 86
Y: 462
Compare the traffic, Threshold=0.5 --black, T=0.1 blue
The vertex number in network
The traffic
T=0.5
T=0.5
T=0.1
T=0.1

Figure A-10: Traffic Analysis for the Sensor Networks with Threshold 0.1 (Blue Line) and
Threshold 0.5 (--Black Line)
0 10 20 30 40 50 60 70 80 90 100
200
250
300
350
400
450
500
550
600
650
X: 88
Y: 210
Compare the Distant squre, Threshold=0.5 --black, T=0.1 blue
The vertex number in network
The distance square
X: 91
Y: 270

Figure A-11: Inertia of the Sensor Networks with Threshold 0.1 (Blue Line) and Threshold
0.5 (--Black Line)

108
4 Waterflooding in Petroleum Engineering
Waterflooding is a method of secondary recovery in which water is injected into the
reservoir formation to displace residual oil. The water from injection wells physically
sweeps the displaced oil to adjacent production wells as shown in Figure 1-2. Since with
current technology, an average of 70% of oil cannot be pumped out from the oil fields in
the primary production, the ever-increasing demand for petroleum worldwide makes
waterflooding a more and more popular research topic in petroleum engineering. The use
of smart wells
3
and other formation evaluation technologies (such as sonic and MR scanner,
etc.) are also gaining popularity in the industry for better understanding of the reservoir
environment such as geological data.
Waterflooding is a typical optimization problem. The optimization objective is to
maximize the net present value with proper water-oil ratio by adjusting a set of control
parameters such as production wells’ pressure and water flow injection rates. To solve this
optimization problem, two kinds of approaches are currently being used: one is using
optimal control and non-linear programming, which require the explicit knowledge of the
underlying equations for their computational solutions; another approach is using
estimation theory such as the Extended Kalman Filter (EKF), neural network, and decision
tree modeling to forecast the injector-producer relationships between multiple injectors and
producers based on measured production and injection rates.
Because of the large and complicated nature of reservoir models with large number of
unknowns and non-linear constraints, the first approach is time consuming and unpractical.

3
The smart wells are typically wells that are equipped with downhole chokes and other measuring instruments.

109
The second approach is assuming the reservoir is a black box and only considers the
correlation and time lag between input (water injection rates) and output (production rates).
Besides that, the advantage of knowing something about the reservoir environment hasn’t
been fully utilized.
The tradeoff between the two approaches is that we know something about the system
(“there is reservoir in here”) but we only have a rough picture about it. We can not expect
to solve the whole optimization problem only based on this limited information, but we will
not throw away the useful information about the system (such as geological data). The
coarse geometry approach is a potentially promising method to achieve a trade-off in this
situation by only considering the most important global factors.
Specifically, from the estimation theory point of view, a waterflood response is
estimated as a rank correlation between two flow rate changes for all of injector-producer
pairs, as shown in Figure A-12 [Fedenczuk03]. In Figure A-13 [Fedenczuk03], the
correlations (oil, gas, and total fluid responses) and associated time lags are presented in a
composite spine diagram. It shows the fluid communication through the reservoir. In this
spine diagram, green and red lines represent positive and negative correlations respectively.
Solid and dashed lines represent responses with a time lag below and above six months
respectively.

110

Figure A-12: A Idea Lagged Waterflood Response

Figure A-13: Composite Spider Diagram for Oil Responses.
These correlation data in fluid communication can be applied as the primary distance
measurement of coarse geometry. The curvature and inertia can be developed based on this
data. The water is injected through the injector well, forced to propagate through pore
spaces and sweeps the oil towards the producing wells. Normally, the percentage of water
in the produced fluids steadily increases. Once the percentage is too high to economically
produce crude oil, waterflooding has to be stopped. This is because the injected water finds
its way through conductive fractures and high permeability zones within the reservoir. This

111
premature breakdown mostly occurs in highly heterogeneous reservoirs. As a result, many
water injectors do not usually achieve improved sweep efficiencies and a lot of the oil is
by-passing. It is important to impose a suitable pressure or flow rate profile along the
injection wells to prevent this premature breakdown.
From our theorem, we can interpret the above situation as follows: the reservoir is a
fluid network. Every path has a threshold, and the pressure of water or oil has to exceed
this threshold for fluid to pass through the path. Once the pressure is too high, the network
will be positively curved, as the fluid makes it way through the great many tiny fractures.
But this causes a high percentage of water in the produced fluids. The Strategy that utilizes
our paradigm is: i) apply proper pressure of water to prevent the emergence of a positively
curved fluid network; ii) place the production wells on the minimum inertia position of the
fluid network to get maximum congestion/congregation of oil;. Of course, more detailed
geological data will help us to refine the network model. For example, Figure A-14 shows a
2D view of waterflooding in fluid network.

Figure A-14: An Idea Waterflooding in the Reservoir (Left: Before the Waterflooding;
Right: after the Waterflooding; Blue Represents Water and White Represents Oil)

Abstract (if available)

Abstract In this thesis, the fundamental experimental observation is that, on the Internet and other networks, traffic seems to concentrate quite heavily on some very small subsets. The main result in here is that this phenomenon is not, in general, related to the popular "heavytailed" phenomenon, but is a consequence of the negative curvature of the network. The mathematical analysis and simulation results confirm this striking traffic pattern specific to negatively curved networks from both the theoretical and practical points of view. Furthermore, this thesis addresses another fundamental question: if congestion does not necessarily occur at vertices with high degree, nor at the so-called highly connected "core," then what are the congestion points? It is shown that the congestion points are the points relative to which the network has low moment of inertia. That single point relative to which the network has a well-defined minimum is referred to as the centroid, the point through which most of the traffic transits. Probably the most important result as far as protocol design is concerned is that load balancing can be achieved by a routing table designed on the basis of a virtual network in which the link weights have been adjusted to correct the curvature from negative to positive. The latter mathematical technique is reminiscent of the Yamabe flow and the Poincaré conjecture. The practical implementation of this concept on the ns-2 network simulator indicates a significant reduction of the congestion.

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Asset Metadata

Creator Lou, Mingji (author)

Core Title Traffic pattern in negatively curved network

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/20/2008

Defense Date 05/09/2008

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

Tag coarse geometry,Communication,congestion,control theorem,curvature,Internet,networking,OAI-PMH Harvest,Traffic

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jonckheere, Edmond A. (committee chair), Baryshnikov, Yuliy (committee member), Bonahon, Francis (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email loumingji@yahoo.com,mingjilou@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1682

Unique identifier UC1218293

Identifier etd-Lou-2373 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-116627 (legacy record id),usctheses-m1682 (legacy record id)

Legacy Identifier etd-Lou-2373.pdf

Dmrecord 116627

Document Type Dissertation

Rights Lou, Mingji

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu