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此书稿尚非最后版本，切勿传阅。

Introduction to

Complex Networks:

Models, Structures and Dynamics

Guanrong Chen, Xiaofan Wang, Xiang Li

copyrighted by the authors

1 January 2012

Table of Contents

Preface

Part I Fundamental Theory

Chapter 1 Introduction ………………………………………………………….……..1

1.1 Some Background and Motivation ………………………………………….……...1

1.2 A Brief History of Complex Network Research …………………………………...4

1.2.1 The Königsburg Seven-Bridge Problem …………………………………….. 4

1.2.2 Random Graph Theory ………………………………………………….…… 5

1.2.3 Small-World Experiment ……………………………………………………. 7

1.2.4 Strength of Weak Ties ………………………………………………………. 9

1.2.5 New Era of Complex-Network Studies ……………………………………. 10

1.3 Some Basic Concepts ……………………………………………………………. 12

1.3.1 Graph Representation of Networks ………………………………………… 12

1.3.2 Average Path Length ……………………………………………………….. 13

1.3.3 Clustering Coefficient ………………………………………………………. 14

1.3.4 Degree and Degree Distribution ……………………………………………. 16

1.3.5 Statistical Properties of Some Real-World Complex Networks ……………. 20

Problems ………………………………………………………………………………22

References …………………………………………………………………………….25

Chapter 2 A Brief Introduction to Graph Theory …………………………………… 27

2.1 What is a Graph? …………………………………………………………………. 27

2.2 Notation, Definitions and Preliminaries ….………………………………………. 29

2.3 Eulerian and Hamiltonian Graphs ………………………………………………... 38

2.3.1 Eulerian graphs ……………………………………………………………... 38

2.3.2 Hamiltonian graphs …………………………………………………………. 40

2.4 The Chinese Postman Problem …………………………………………………… 41

2.5 The Shortest Path Length Problem ……………………………………………….. 44

2.6 Trees ………………………………………………………………………………. 46

2.7 The Minimum Connector Problem ……………………………………………….. 48

2.8 Plane Graphs and Planar Graphs …………………………………………………. 49

2.9 Euler Formula for Plane Graphs ………………………………………………….. 51

2.10 Directed Graphs …………………………………………………………………. 52

Problems ……………………………………………………………………………… 57

References ……………………………………………………………………………. 65

Chapter 3 Network Topologies: Basic Models and Properties ……………………… 66

3.1 Introduction ………………………………………………………………………. 66

3.2 Regular Networks ……………………………………………………………….…66

3.3 Random-Graph Networks ………………………………………………………… 69

3.4 Small-World Network Models …………………………………………………… 71

3.4.1 The WS Small-World Network Model …………………………………….. 72

3.4.2 The NW Small-World Network Model ……………………………………. 72

3.4.3 Statistical Properties of Small-World Network Models …………………… 73

3.5 The Navigable Small-World Network Model …………………………………… 77

3.6 Scale-Free Network Models ……………………………………………………... 79

3.6.1 The BA Scale-Free Network Model ……………………………………….. 80

3.6.2 Robustness versus Fragility ………………………………………………... 83

3.6.3 Modified BA Models ………………………………………………………. 87

3.6.4 A Simple Model with Power-Law Degree Distribution ………………….... 91

3.6.5 Local-World and Multi-Local-World Network Models …………………….92

Problems ………………………………………………………………………….…. 101

References ………………………………………………………………………….. 105

Part II Applications - Selected Topics

Chapter 4 Internet: Topology and Modeling ……………………………………….. 109

4.1 Introduction ……………………………………………………………………… 109

4.2 Topological Properties of the Internet ……………………………….……………111

4.2.1 Power-Law Node-Degree Distributions ……………………….……………112

4.2.2 Hierarchical Structures ……………………………………….……………..115

4.2.3 Rich-Club Structure ………………………………………….……………..117

4.2.4 Disassortative Property ……………………………………………………. 118

4.2.5 Coreness and Betweenness …………………………………………………120

4.2.6 Growth of the Internet ………………………………………………………125

4.2.7 Router-Level Internet Topology ………………………………….…...…….127

4.2.8 Geographic Layout of the Internet ………………………………………….128

4.3 Random-Graph Network Topology Generator …………………………………...131

4.4 Structural Network Topology Generators ………………………………………...131

4.4.1 Tiers Topology Generator …………………………………………………. 132

4.4.2 Transit-Stub Topology Generator …………………………………………. 134

4.5 Connectivity-Based Network Topology Generators …………………………….. 137

4.5.1 Inet ………………………………………………………………………… 137

4.5.2 BRITE Model ……………………………………………………………… 139

4.5.3 GLP Model ………………………………………………………………… 141

4.5.4 PFP Model …………………………………………………………………. 142

4.5.5 T

ANG

Model …………………………………………………………………144

4.6 Multi-Local World Model ………………………………………………………...146

4.6.1 Theoretical Considerations …………………………………………………146

4.6.2 Numerical Results with Comparison ……………………………………….149

4.6.3 Performance Comparison ………………………………………………….. 158

4.7 HOT Model ………………………………………………………………………. 160

4.8 Dynamical Behaviors of the Internet Topological Characteristics ………………. 163

References ……………………………………………………………………………. 166

Chapter 5 Spreading Dynamics ………………………………………………………170

5.1 Introduction ………………………………………………………………………..170

5.2 Epidemic Threshold Theory …………………………………………………….…171

5.2.1 Epidemic Models ………………………………………………………….…171

5.2.2 Epidemic Thresholds on Homogenous Networks ………………………….. 173

5.2.3 Statistical Data Analysis ………………………………………….………….174

5.2.4 Epidemic Thresholds on Scale-Free Networks ………………..…….……… 175

5.2.5 Epidemic Thresholds on BA Scale-Free Networks …………………………176

5.2.6 Epidemic Thresholds on Finite-Sized Scale-Free Networks ………………..178

5.2.7 Epidemic Thresholds on Correlated Networks …………………………….. 179

5.2.8 Epidemic Thresholds on Some Generalized Scale-Free Networks …………180

5.2.9 SIR Model of Epidemic Spreading …………………………………………..181

5.3 Immunization on Complex Networks ……………………………………………. 183

5.3.1 Random Immunization ……………………………………………………... 183

5.3.2 Targeted Immunization ……………………………………………………...184

5.3.3 Acquaintance Immunization ……………………………………………...…186

5.4 Computer Virus Spreading over the Internet ……………………………………...186

5.4.1 Random Constant Spread Model of the Code-Red Worm …………………..187

5.4.2 A Compartment-Based Model of Computer Worms ………………………. 188

5.4.3 Spreading Models of Email Viruses …………………………………….….. 191

5.4.4 Effects of Computer Virus on Network Topologies ……………………….. 196

5.5 Other Spreading Phenomena on Complex Networks ……………………………..198

5.5.1 Rumors Spreading over Social Networks ………………………………….. 199

5.5.2 Some Generalized Models of Spreading Dynamics …………………………200

5.6 Spreading Dynamics on Complex Networks …………………………………….. 202

References ……………………………………………………………………………. 203

Chapter 6 Cascading Reactions on Networks ………………………………………. 205

6.1 Introduction ………………………………………………………………………. 205

6.2 Dynamic Cascading Failures: Models and Analyses …………………………….. 205

6.2.1 Models Based on Node Dynamics ………………………………….……… 205

6.2.2 Models Based on Edge Dynamics …………………………………………. 211

6.2.3 Hybrid Models Based on Node and Edge Dynamics ……………………… 214

6.2.4 Binary Influence Model …………………………………………………… 217

6.2.5 Sand-Pile Model …………………………………………………………… 219

6.2.6 OPA Model ………………………………………………………………… 221

6.2.7 CASADE Model …………………………………………………………… 225

6.2.8 Other Models …………………………………………………………….… 228

6.3 Cascading Failures in Coupled Map Lattices ……………………………………. 229

6.3.1 Cascading Failure Model Based on CMLs ………………………………… 229

6.3.2 Cascading Failures on Typical Coupling Lattices …………………………. 230

6.4 Cascading Failures of Interdependent Networks ……………………………….... 240

References ………………………………………………………………………….… 242

Chapter 7 Human Opinion Dynamics ………………………………………………. 244

7.1 Introduction …………………………………………………………………….…244

7.2 Social Network Topologies and Sociodynamics ……………………………….... 245

7.3 Social Opinion Formation ……………………………………………………….. 247

7.3.1 Voter Model ……………………………………………………………….. 247

7.3.2 Galam Majority-Rule Model ………………………………………………. 249

7.3.3 Latané Social Impact Theory ………………………………………….…… 256

7.3.4 Sznajd Model ………………………………………………………….…… 257

7.3.5 Virtual Social Game on the Internet …………………………………….…. 259

7.3.6 Online Social Opinion Formation ………………………………………….. 261

7.4 Bounded Confidence Models …………………………………………………….. 263

References ……………………………………………………………………………. 266

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