Complex Webs: Anticipating the Improbable

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 The myth of normalcy......Page 13
1.1.1 Knowledge, information and uncertainty......Page 20
1.1.2 Maximum entropy......Page 26
1.2 Empirical laws......Page 33
1.3 Why people forecasts are usually wrong......Page 46
1.3.1 Power laws, prediction and failure......Page 50
1.4 Recap......Page 53
1.4 The divine proportion......Page 54
References......Page 55
2 Webs, trees and branches......Page 57
2.1 Natural trees and scaling......Page 62
2.1.1 Two kinds of scaling......Page 64
2.1.2 Physiologic trees......Page 71
2.2 Physiology in fractal dimensions......Page 77
2.2.1 Fractals......Page 78
2.2.2 Temporal scaling......Page 86
2.2.3 Autocorrelation functions and fractal dimensions......Page 89
2.3 Measuring what we cannot know exactly......Page 91
2.3.1 Inverse power laws of Zipf and Pareto......Page 92
2.3.2 Chaos, noise and probability......Page 97
2.4 Scaling physiologic time series......Page 100
2.4.1 Fractal heartbeats......Page 101
2.4.2 Fractal breaths......Page 104
2.4.3 Fractal steps......Page 106
2.5 Some afterthoughts......Page 110
2.3 The generalized Weierstrass function......Page 111
References......Page 112
3 Mostly linear dynamics......Page 117
3.1 Hamilton's equations......Page 119
3.2 Linear stochastic equations......Page 126
3.2.1 Classical diffusion......Page 130
3.2.2 Linear response theory (LRT)......Page 136
3.2.3 Double-well potential and stochastic resonance......Page 139
3.3 More on classical diffusion......Page 142
3.3.1 The Fokker–Planck equation (FPE)......Page 145
Multiplicative fluctuations......Page 147
3.3.2 The drift-diffusion model (DDM) of decision making......Page 149
3.3.3 First-passage-time distributions......Page 154
3.4 More on statistics......Page 156
3.4.1 Poisson statistics......Page 157
3.4.2 Statistics with memory......Page 159
3.4.3 Inverse power-law correlations......Page 160
3.5.1 Age-specific failure rates......Page 165
3.5.2 Possible forms of failure rates......Page 167
3.5.3 Various Poisson processes......Page 170
3.6 Overview......Page 173
3.2 A Hamiltonian heat bath......Page 174
References......Page 175
4 Random walks and chaos......Page 178
4.1 Random walks......Page 180
The Bernoulli sequence......Page 181
4.1.2 Poisson webs......Page 186
4.2 Fractal random walks......Page 189
4.2.1 The Weierstrass walk......Page 190
4.2.2 Lévy random flights......Page 192
4.2.3 Fractional operator walk......Page 195
Financial time series......Page 197
4.3 Maps, chaos and strange attractors......Page 200
4.3.1 The logistic equation......Page 202
4.3.2 Strange attractors......Page 210
4.4 Integrable and non-integrable Hamiltonians......Page 214
4.4.1 The surface of section......Page 217
4.4.2 Chaos and linear response theory......Page 220
The prototype booster......Page 225
Response to a weak external perturbation......Page 227
4.5 Précis......Page 231
4.3 Anomalous diffusion......Page 232
References......Page 233
5 Non-analytic dynamics......Page 236
5.1 Transition to fractional dynamics......Page 237
5.1.1 Fractional calculus......Page 240
Fractal function evolution......Page 242
5.1.2 Deterministic fractional dynamics......Page 243
5.2.1 The fractional Poisson distribution......Page 246
5.2.2 The Weibull distribution......Page 248
5.3.1 The fractional Langevin equation......Page 249
5.3.2 Multiplicative random force......Page 251
Properties of multifractals......Page 253
Migraine and cerebral blood flow......Page 254
Stride-interval multifractality......Page 257
5.4.1 Random fields......Page 259
5.4.2 Continuous-time random walks (CTRWs)......Page 262
5.4.3 Infinite moments......Page 266
5.4.4 Scaling solutions......Page 269
5.5 Reiteration......Page 270
References......Page 271
6 A brief recent history of webs......Page 274
6.1 Web growth......Page 276
6.1.1 Scale-free webs......Page 277
6.1.2 Communication webs......Page 280
6.2 Graphic properties of webs......Page 287
Some definitions......Page 289
The Poisson distribution of links......Page 291
Giant clusters......Page 293
The length of the random web......Page 294
The clustering coefficient......Page 295
6.2.2 Real webs......Page 298
Beyond the model of preferential attachment......Page 303
6.2.3 Web failures......Page 306
6.3.1 Hierarchal webs......Page 309
6.3.2 From small to ultra-small networks......Page 313
6.4 Reflections......Page 315
6.3 Fractal curves......Page 316
References......Page 317
7 Dynamics of chance......Page 319
7.1 The master equation......Page 320
7.1.1 Discrete dynamics......Page 321
7.1.2 A synchronized web of two-level nodes......Page 322
7.1.3 Stochastic approximation to a synchronized web......Page 326
7.1.4 The Yule process......Page 329
7.2.1 From crucial events to the GME......Page 331
7.2.2 Relaxation of the two-level GME......Page 334
7.2.3 Perturbing the GME......Page 335
7.2.4 Towards a new LRT......Page 338
7.2.5 Stochastic resonance and conventional LRT......Page 339
7.3 The Universality of LRT......Page 343
7.3.1 The Hamiltonian approach to LRT......Page 344
7.3.2 Applying the formalism......Page 346
7.3.3 The experimental approach......Page 348
7.3.4 Theoretical interpretation......Page 352
7.4 The statistical habituation model (SHM)......Page 356
7.4.1 Simple stimulus and renewal theory......Page 358
Connection with power spectra......Page 359
7.4.2 Information resonance and dishabituation......Page 360
7.4.3 Discussion......Page 361
7.5 Summary......Page 363
7.4 Habituation to arbitrary stimulus......Page 364
References......Page 365
8 Synopsis......Page 369
References......Page 375
Index......Page 377