Statistical Mechanics of Complex Networks

front-matter......Page 1
1......Page 12
2.1 Introduction......Page 14
2.2.1 The Degree Distribution......Page 15
2.2.2 Order Distribution......Page 18
2.2.3 Degree Correlations......Page 19
2.2.4 Global Properties......Page 20
2.3.1 Role of Finiteness......Page 22
2.3.2 Extremes and Lead Changes......Page 24
2.4.1 In.nite-Order Percolation Transition......Page 27
2.4.2 Non-universal Degree Distribution......Page 30
2.5 Outlook......Page 32
3.1.1 Percolation Threshold......Page 34
3.1.2 Generating Functions......Page 38
3.2 Directed Graphs......Page 39
3.2.1 Structure......Page 40
3.2.2 Percolation Threshold......Page 41
3.2.3 Critical Exponents......Page 44
3.2.4 Summary......Page 47
3.3 Spatially Embedded Scale-Free Graphs......Page 48
3.3.1 Model De.nition......Page 49
3.3.2 Summary......Page 54
4.1 Introduction......Page 57
4.2 Hierarchical Network Model......Page 60
4.3 Hierarchical Organization in Non-biological Networks......Page 63
4.4 Hierarchy in Metabolic Networks and the Functional Organization of Escherichia Coli......Page 66
4.5 Stochastic Model and Universality......Page 71
4.6 Discussion and Outlook......Page 73
5.1 Introduction......Page 77
5.2 Community Structure......Page 78
5.2.1 Edge Betweenness and Community Detection......Page 80
5.2.2 Examples......Page 82
5.3 Origins of Community Structure and Assortative Mixing......Page 84
5.4 Other Types of Assortative Mixing......Page 89
5.4.1 Mixing by Vertex Degree......Page 90
5.5 Conclusions......Page 95
6.1 The Meaning of Acceleration in Networks......Page 99
6.2.1 What Types of Degree Distribution Can We Have?......Page 102
6.2.2 The Most Interesting Case: Power-Law Degree Distribution......Page 104
6.3 General Relations for the Accelerated Growth......Page 109
6.4 Scaling Relations for Accelerated Growth......Page 111
6.5.1 Model for γ < 2......Page 112
6.5.2 Model for γ > 2......Page 113
6.6 One Practical Example: The Word Web......Page 116
6.7 Conclusions......Page 120
7.1 Introduction......Page 125
7.2 Network Optimization......Page 126
7.3 The Optimization Algorithm......Page 127
7.4 Optimal Degree Distributions......Page 130
7.5 Discussion......Page 133
Appendix......Page 134
8.1 Introduction......Page 138
8.2 Correlated Complex Networks......Page 139
8.2.1 Assortative and Disassortative Mixing......Page 140
8.2.2 Degree Detailed Balance Condition......Page 141
8.3.1 Uncorrelated Homogeneous Networks......Page 144
8.3.2 Uncorrelated Complex Networks......Page 145
8.3.3 Correlated Complex Networks......Page 146
8.3.4 Correlated Scale-Free Networks......Page 147
8.4 The SIR Model......Page 149
8.4.1 Uncorrelated Homogeneous Networks......Page 150
8.4.2 Uncorrelated Complex Networks......Page 151
8.4.3 Correlated Complex Networks......Page 153
8.5 Conclusions......Page 155
9.1 Introduction......Page 159
9.2.1 Ecological Properties......Page 161
9.2.3 Degree Distribution......Page 162
9.3 Spanning Tree Analysis......Page 163
9.3.2 Allometric Scaling Relations......Page 164
9.3.3 E.ciency of Empirical Food Webs......Page 165
9.3.4 Stability under Species Removal......Page 167
9.4 The Webworld Model......Page 168
9.4.2 Initial State of the Model......Page 169
9.4.3 Evolution of the Model......Page 171
9.4.4 Further Comments on the Model......Page 175
9.5 Discussion and Conclusion......Page 176
10.1 Introduction......Page 178
10.2 Recent Advances on Scale-Free Social Networks......Page 179
10.4.1 Classi.cation of Real Networks......Page 180
10.4.4 Optimizing the Stability of Threatened Networks......Page 181
10.5 Possible Contributions of Social Network Research......Page 182
10.6 Discussion......Page 183
11.1 Introduction......Page 186
11.2 Search in Complex Networks......Page 187
11.3 Load and Congestion in Complex Networks......Page 189
11.4.1 Description of the Model......Page 190
11.4.2 Congestion and Network Capacity......Page 192
11.5 Analytical Results for Hierarchical Lattices......Page 193
11.6 Optimization in Model Networks......Page 195
11.6.1 Network Topology......Page 196
11.6.2 Communication Model and Search Algorithm......Page 197
11.6.3 Results......Page 198
11.7 Optimization in a General Framework......Page 201
11.8 Summary......Page 204
12.1 Introduction......Page 206
12.2 The Hodgkin-Huxley Model......Page 207
12.3 Stochastic Version of the Hodgkin-Huxley Model......Page 210
12.3.1 Quantifying Channel Noise......Page 211
12.3.2 Stochastic Resonance......Page 213
12.4 Conclusions......Page 215