Stefan Bornholdt and Heinz Georg Schuster
HANDBOOK OF GRAPHS AND NETWORKS
From the Genome to the Internet
Wiley-VCH (2003)
Contents in pdf
Preface ..................................................................................................................... V
List of contributors ..............................................................................................XIV
1 Mathematical results on scale-free random graphs (Béla Bollobás and Oliver M.
Riordan)
1.1 Introduction ........................................................................................................1
1.2 Classical models of random graphs ...................................................................2
1.3 Results for classical random graphs .................................................................. 4
1.4 The Watts-Strogatz 'small-world' model. ........................................................... 5
1.5 Scale-free models ............................................................................................... 6
1.6 The Barabási-Albert model ................................................................................7
1.7 The LCD model and Gm(n)..................................................................................9
1.8 The Buckley-Osthus model ..............................................................................11
1.9 The copying model........................................................................................... 12
1.10 The Cooper-Frieze model ..............................................................................13
1.11 Directed scale-free graphs ..............................................................................15
1.12 Clustering coefficient and small subgraphs ...................................................17
1.13 Pairings on [0,1] and the diameter of the LCD model ...................................22
1.14 Robustness and vulnerability .........................................................................24
1.15 The case [0,1] plane-oriented recursive trees .................................................27
1.16 Conclusion .....................................................................................................32
References ..............................................................................................................32
2 Random graphs as models of networks (Mark E. J. Newman)
2.1 Introduction ......................................................................................................35
2.2 Random graphs with specified degree distributions ........................................40
2.3 Probability generating functions ......................................................................45
2.3.1 Properties of generating functions ................................................................ 46
2.3.2 Examples .......................................................................................................46
2.4 Properties of undirected graphs ........................................................................47
2.4.1 Distribution of component sizes. ..................................................................47
2.4.2 Mean component size....................................................................................50
2.4.3 Above the phase transition. ...........................................................................51
2.5 Properties of directed graphs................................................................................53
2.5.1 Generating functions .........................................................................................54
2.5.2 Results ...............................................................................................................54
2.6 Networks with clustering .....................................................................................56
2.7 Models defined on random graphs .......................................................................58
2.7.1 Network resilience ............................................................................................58
2.7.2 Epidemiology ....................................................................................................61
2.7.3 The SIR model ..................................................................................................62
2.7.4 Solution of the SIR model .................................................................................63
2.8 Summary ..............................................................................................................65
References. .................................................................................................................65
3 Emergence of scaling in complex networks (Albert-László Barabási)
3.1 Introduction ..........................................................................................................69
3.2 Network models ...................................................................................................70
3.2.1 Random networks ............................................................................................. 70
3.2.2 Scale-free networks ...........................................................................................70
3.2.3 Scale-free model ............................................................................................... 73
3.3 Fitness model and Bose-Einstein condensation ...................................................75
3.4 The Achilles' Heel of complex networks ............................................................. 76
3.5 A deterministic scale-free model .........................................................................79
3.6 Outlook.................................................................................................................81
3.7 Acknowledgments ................................................................................................ 82
References ..................................................................................................................82
4 Structural properties of scale-free networks (Reuven Cohen, Shlomo Havlin, and Daniel
ben-Avraham)
4.1 Introduction ..........................................................................................................85
4.1.1 Random graphs .................................................................................................85
4.1.2 Scale-free networks ...........................................................................................86
4.2 Small and Ultra-small worlds ..............................................................................87
4.2.1 Diameter of scale-free networks .......................................................................88
4.2.2 Minimal graphs and lower bound .....................................................................88
4.2.3 The general case of random scale-free networks ..............................................89
4.3 Percolation ...........................................................................................................92
4.3.1 Random breakdown ..........................................................................................92
4.3.2 Percolation critical threshold ............................................................................93
4.3.3 Generating functions .........................................................................................95
4.3.4 Intentional attack ............................................................................................... 96
4.3.5 Critical exponents ............................................................................................. 97
4.3.6 Fractal dimension ........................................................................................... 100
4.4 Percolation in directed networks ....................................................................... 101
4.4.1 Threshold ....................................................................................................... 102
4.4.2 Critical exponents .......................................................................................... 103
4.5 Efficient immunization strategies ..................................................................... 104
4.5.1 Acquaintance immunization .......................................................................... 105
4.6 Summary and outlook ....................................................................................... 106
References ............................................................................................................... 107
5 Epidemics and immunization in scale-free networks (Romualdo Pastor-Satorras and
Alessandro hespignani)
5.1 Introduction ....................................................................................................... 111
5.2 Computers and epidemiology ........................................................................... 112
5.3 Epidemic spreading in homogeneous networks ................................................ 114
5.4 Real data analysis .............................................................................................. 116
5.5 Epidemic spreading in scale-free networks ...................................................... 118
5.5.1 Analytic solution for the Barabási-Albert network ........................................ 119
5.5.2 Finite size scale-free networks ....................................................................... 122
5.6 Immunization of scale-free networks ................................................................ 123
5.6.1 Uniform immunization ................................................................................... 124
5.6.2 Targeted immunization .................................................................................. 125
5.7 Conclusions ....................................................................................................... 127
References ............................................................................................................... 128
6 Cells and genes as networks in nematode development and evolution (Ralf J. Sornmer)
6.1 Introduction ....................................................................................................... 131
6.2 Nematode developmental biology: studying processes at a cellular level ........ 132
6.3 Nematode Vulva formation as a case study ...................................................... 132
6.4 Nematode collections ........................................................................................ 136
6.5 Cellular networks: how cells change their function .......................................... 136
6.5.1 Evolution of vulva position ............................................................................ 136
6.5.2 Evolution of vulval cell fate specification ..................................................... 136
6.6 Genetic networks: how genes change their function ........................................ 139
6.6.1 Evolution of lin-39 function ........................................................................... 139
6.6.2 Evolution of mab-5 function .......................................................................... 141
6.7 Conclusion ........................................................................................................ 142
References ............................................................................................................... 142
7 Complex networks in genomics and proteomics (Ricard V. Solé and Romualdo PastorSatorras)
7.1 Introduction ....................................................................................................... 145
7.2 Cellular networks .............................................................................................. 148
7.2.1 Two-gene networks ........................................................................................ 149
7.2.2 Random networks .......................................................................................... 150
7.3 Three interconnected levels of cellular nets ...................................................... 153
7.4 Small world graphs and scale-free nets............................................................. 154
7.5 Scale-free proteomes: gene duplication models ................................................ 157
7.5.1 Mean-field rate equation for the average connectivity .................................. 158
7.5.2 Rate equarion for the node distribution nk ..................................................... 159
7.5.3 Numerical simulations ................................................................................... 162
7.6 Discussion ......................................................................................................... 164
References. .............................................................................................................. 164
8 Correlation profiles and motifs in complex networks (Sergei Maslov, Kim Sneppen, and
Uri Alon)
8.1 Introduction ....................................................................................................... 168
8.2 Randomization algorithm: Constructing the proper null model ....................... 172
8.3 Correlation profiles: Yeast molecular networks and the Internet ..................... 177
8.4 Network motifs: Transcriptional regulation in E. coli ...................................... 189
8.5 Discussion: What it may all mean? ................................................................... 194
References ............................................................................................................... 196
9 Theory of interacting neural networks (Wolf'gang Kinzel)
9.1 Introduction ....................................................................................................... 199
9.2 On-line training ................................................................................................. 200
9.3 Generalisation ................................................................................................... 201
9.4 Time series prediction and generation .............................................................. 203
9.5 Self-interaction .................................................................................................. 206
9.6 Agents competing in a closed market ............................................................... 207
9.7 Synchronisation by mutual learning ................................................................. 208
9.8 Cryptography .................................................................................................... 210
9.9 Conclusions ....................................................................................................... 213
References ............................................................................................................... 216
10 Modelling food webs (Barbara Drossel and Alan J. McKane)
10.1 Introduction ..................................................................................................... 218
10.2 Basic properties of food webs ......................................................................... 221
10.3 Static models ................................................................................................... 226
10.4 Dynamic models ............................................................................................. 227
10.4.1 Two-species models ..................................................................................... 228
10.4.2 Generalized dynamical equations ................................................................ 230
10.4.3 The complexity-stability debate ................................................................... 232
10.5 Assembly models and evolutionary models .................................................... 235
10.5.1 Toy models ................................................................................................... 235
10.5.2 Species assembly models ............................................................................. 236
10.5.3 Evolutionary models .................................................................................... 238
10.6 Conclusions ..................................................................................................... 241
References ............................................................................................................... 242
11 Traffic networks (Kai Nagel)
1l.l Introduction ....................................................................................................... 248
11.2 Dynamics on networks .................................................................................... 250
11.2.1 The four step process and static assignment ................................................ 250
11.2.2 Simple link dynamics and the queue model ................................................. 252
11.2.3 Virtual reality micro-simulations ................................................................. 253
11.2.4 CA implementations of virtual reality micro-simulations ........................... 255
11.2.5 Traffic in networks ....................................................................................... 258
11.3 Particles are intelligent .................................................................................... 260
11.3.1 Route generation .......................................................................................... 260
11.3.2 Activity generation ....................................................................................... 261
11.3.3 Housing, land use, freight, life style, et al .................................................... 261
11.3.4 Day-to-day learning, feedback, and relaxation ............................................ 261
11.3.5 Within-day re-planning ................................................................................ 262
11.3.6 Individualization of knowledge ................................................................... 263
11.3.7 State of the art .............................................................................................. 263
11.4 Distributed computing and the network of interactions .................................. 264
11.4.1 Distributed computing of the traffic micro-simulation ................................ 265
11.4.2 Distributed computing of plans generation .................................................. 267
11.5 Outlook: Dynamics of networks ..................................................................... 268
11.6 Conclusion ...................................................................................................... 268
References ............................................................................................................... 269
12 Economic networks (Alan Kirman)
12.1 Introduction ..................................................................................................... 273
12.2 Economics and sociology................................................................................ 274
12.3 The economic consequences of networks ....................................................... 275
12.4 Fixed network: stochastic interaction ............................................................. 278
12.5 Random graphs and networks ......................................................................... 280
12.6 Emerging networks ......................................................................................... 281
12.7 The strategic formation of networks ............................................................... 282
12.8 Emerging random graphs ................................................................................ 283
12.9 The identification problem .............................................................................. 291
12.10 Conclusion .................................................................................................... 292
References ............................................................................................................... 293
13 Local search in unstructured networks (Lada A. Adamic, Rajan M. Lukose, Bernardo A.
Huberman)
13.1 Introduction. .................................................................................................... 295
13.2 Search in power-law random graphs ............................................................... 297
13.2.1 Intuition ........................................................................................................ 297
13.2.2 Random walk search. ................................................................................... 298
13.2.3 Search utilizing high degree nodes .............................................................. 301
13.3 Simulation ....................................................................................................... 303
13.4 Comparison with Poisson distributed graphs .................................................. 306
13.5 Gnutella ........................................................................................................... 308
13.6 Path finding ..................................................................................................... 310
13.7 Shortening the shortest path ............................................................................ 312
13.7.1 Iterative deepening ....................................................................................... 313
13.8 Adaptive search ............................................................................................... 314
13.9 Conclusion ...................................................................................................... 315
References ............................................................................................................... 316
14 Accelerated growth of networks (Sergei N. Dorogovtsev and Jose F. F. Mendes)
14.1 Acceleration .................................................................................................... 318
14.2 Reasons for acceleration ................................................................................. 321
14.3 Degree distributions of networks .................................................................... 321
14.3.1 Types of degree distribution ........................................................................ 321
14.3.2 Power-law degree distribution ..................................................................... 324
14.4 General relations for accelerated growth ........................................................ 326
14.5 Scaling relations for accelerated growth ......................................................... 328
14.6 Degree distributions produced by acceleration ............................................... 329
14.6.1 Model for  < 2 ............................................................................................. 329
14.6.2 Model for > 2 ............................................................................................. 330
14.6.3 Dynamically induced accelerated growth .................................................... 330
14.6.4 Partial copying of edges and multifractality ................................................ 330
14.7 Evolution of the Word Web ............................................................................ 331
14.8 Wealth distribution in evolving societies ........................................................ 336
14.8.1 Stable (stagnating) societies ......................................................................... 337
14.8.2 Developing and degrading societies ............................................................ 337
References ............................................................................................................... 339
15 Social percolators and self organized criticality (Gérard Weisbuch and Sorin Solomon)
15.1 Introduction ..................................................................................................... 342
15.2 Social percolation ............................................................................................ 343
15.2.1 Simple models .............................................................................................. 343
15.3 Adjustment meta-dynamics............................................................................. 346
15.3.1 Slow adjustment ........................................................................................... 346
15.3.2 Fast adjustment ............................................................................................ 347
15.4 Conclusions ..................................................................................................... 351
References ............................................................................................................... 353
16 Graph theory and the evolution of autocatalytic networks (Sanjay Jain and Sandeep
Krishna)
16.1 Introduction ..................................................................................................... 355
16.2 Graph theory and autocatalytic sets ................................................................ 357
16.2.1 Directed graphs and their adjacency matrices .............................................. 357
16.2.2 Autocatalytic sets ......................................................................................... 363
16.3 A dynamical system on a fixed graph ............................................................. 366
16.3.1 Attractors of equation (16.1) ........................................................................ 368
16.4 Graph dynamics .............................................................................................. 375
16.5 Self Organization ............................................................................................ 377
16.5.1 The random phase ........................................................................................ 378
16.5.2 The growth phase ......................................................................................... 381
16.5.3 The organized phase..................................................................................... 384
16.6 Catastrophes and recoveries in the organized phase ....................................... 384
16.6.1 Catastrophes, core-shifts and a classification of proximate causes ............. 389
16.6.2 Recoveries .................................................................................................... 391
16.6.3 Correlation between graph theoretic nature of perturbation and its short and long
term impact ............................................................................................................. 392
16.7 Concluding remarks ........................................................................................ 392
References ............................................................................................................... 394
Index........................................................................................................................ 396