Scale-Free Networks Complex Webs in Nature

A variety of different social, natural and technological systems can be described by the same mathematical framework. This holds from the Internet to food webs and to boards of company directors. In all these situations a graph of the elements of the system and their interconnections displaysa universal feature. There are only few elements with many connections, and many elements with few connections. This book presents the experimental evidence of these Scale-free networks and provides students and researchers with a corpus of theoretical results and algorithms to analyse andunderstand these features. The content of this book and the exposition makes it a clear textbook for beginners, and a reference book for the experts.

Guido Caldarelli
Associate Professorship (Primo Ricercatore) National Institute for Condensed Matter
Dipartimento di Fisica,
Universita di Roma "La Sapienza"

Definitions and Methodology
Introduction to graphs	p. 7
Graphs, directed graphs, and weighted graphs	p. 10
Trees	p. 18
Vertex correlation, assortativity	p. 23
Hierarchical properties of graphs	p. 29
The properties of scale-free networks	p. 30
Graph structures: communities	p. 34
Introduction	p. 34
Typical subgraphs, motifs	p. 36
Classes of vertices	p. 38
Centrality measures, betweenness, and robustness	p. 39
Clustering detection, modularity	p. 41
Communities in graphs	p. 43
Scale-invariance	p. 60
Geometrical scale-invariance: fractals	p. 62
Measuring the fractal dimension	p. 67
Scale-invariance and power laws	p. 70
Plotting a power law	p. 71
Scale-invariance in natural sciences	p. 74
Scale-invariance in economics and in social sciences	p. 81
The origin of power-law functions	p. 84
Random walk, Laplace equation, and fractals	p. 86
Power laws from minimization principles	p. 92
Multiplicative processes and normal distribution	p. 95
Preferential attachment, the Matthew effect	p. 100
Graph generating models	p. 103
Random graph model	p. 105
The small-world model	p. 108
The Barabasi-Albert model	p. 112
Modifications to the Barabasi-Albert model	p. 114
Copying models	p. 118
Fitness based model	p. 119
Graph from optimization principles	p. 123
Examples
Networks in the cell	p. 131
Basic cell biology	p. 134
Protein-protein interaction network	p. 138
Metabolic pathways	p. 142
Gene regulatory networks	p. 146
Geophysical networks	p. 148
Satellite images and digital elevation models	p. 150
Geometrical scale invariance for river networks	p. 152
Scaling relations for river networks	p. 155
River networks models	p. 157
River networks on Mars' surface	p. 164
Ecological networks	p. 167
Species and evolution	p. 170
Food webs: a very particular case of network	p. 173
Food web quantities	p. 174
Classifications of species	p. 179
Yule process for taxonomies	p. 180
Technological networks: Internet and WWW	p. 185
The Internet protocols	p. 188
The geography of the Internet	p. 193
The autonomous systems	p. 196
The scale-invariance in the Internet	p. 197
The World Wide Web	p. 199
Searching the web	p. 200
Statistical measures of the Web	p. 205
E-mail networks	p. 208
Social and cognitive networks	p. 211
Networks of scientific papers	p. 213
Contact networks	p. 219
Linguistic networks	p. 224
Wikipedia	p. 229
Financial networks	p. 231
Board of directors	p. 233
Stock networks	p. 235
Bank networks	p. 245
The world trade web	p. 248
Appendices
Glossary	p. 253
Graph quantities	p. 256
Basics	p. 256
Different kinds of graphs	p. 257
Paths, cycles, and trees	p. 260
Basic statistics	p. 262
Events and probability	p. 262
Probability densities and distributions	p. 263
Working with statistical distributions	p. 267
Statistical properties of weighted networks	p. 272
Matrices and eigenvectors	p. 275
Population dynamics	p. 277
Population dynamics	p. 277
Bibliography	p. 283
Index	p. 307
Table of Contents provided by Ingram. All Rights Reserved.

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Scale-Free Networks Complex Webs in Nature and Technology

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