This monograph is a comprehensive and cohesive exposition of power-law statistics. Following a bottom-up construction from a foundational bedrock – the power Poisson process – this monograph presents a unified study of an assortment of power-law statistics including: Pareto laws, Zipf laws, Weibull and Fréchet laws, power Lorenz curves, Lévy laws, power Newcomb-Benford laws, sub-diffusion and super-diffusion, and 1/f and flicker noises. The bedrock power Poisson process, as well as the assortment of power-law statistics, are investigated via diverse perspectives: structural, stochastic, fractal, dynamical, and socioeconomic. This monograph is poised to serve researchers and practitioners – from various fields of science and engineering – that are engaged in analyses of power-law statistics.
Author(s): Iddo Eliazar
Series: Understanding Complex Systems
Publisher: Springer
Year: 2020
Language: English
Pages: 209
Tags: Statistical Physics And Dynamical Systems
Preface......Page 8
Acknowledgements......Page 9
Contents......Page 11
Notation......Page 16
1 Introduction......Page 18
References......Page 24
2.1 Geometric Evolution......Page 29
2.3 Brownian Motions......Page 30
2.4 Gaussian Motions......Page 31
2.5 Gaussian Mean and Variance......Page 32
2.6 Gaussian Stationary Velocities......Page 33
2.7 Poisson-Process Limit......Page 34
2.8 Outlook......Page 35
2.9 Notes......Page 36
2.10.1 Equation (2.8)......Page 38
2.10.2 Equation (2.13)......Page 39
2.10.3 A General Poisson-Process Limit-Law......Page 40
2.10.4 The Power Poisson-Process Limit-Law......Page 42
References......Page 43
3.1 The Poisson Law......Page 46
3.2 Framework......Page 47
3.3.1 Normal Limit for Poisson Random Variables......Page 49
3.3.2 Chernoff Bounds for Poisson Random Variables......Page 50
References......Page 52
4.1 Mean Behavior......Page 54
4.2 Asymptotic Behavior......Page 55
4.3 Log-Log Behavior......Page 56
4.4 Pareto Laws......Page 57
4.5 Pareto Scaling......Page 58
4.7 Weibull Laws......Page 60
4.8 Outlook......Page 62
References......Page 63
5 Hazard Rates......Page 64
References......Page 66
6 Lindy's Law......Page 67
References......Page 74
7.1 Simulation......Page 75
7.2 Statistics......Page 76
7.3 Asymptotic Behavior......Page 78
7.4 Zipf Laws......Page 79
7.5 Ratios......Page 80
7.6 Log-Ratios......Page 81
7.7 Forward Motion......Page 82
7.8 Backward Motion......Page 83
7.9 Outlook......Page 84
7.10.1 Simulation......Page 85
7.10.2 Equations(7.3) and (7.4)......Page 86
7.10.3 Equations(7.9) and (7.11)......Page 87
7.10.4 Equations(7.12) and (7.14)......Page 90
7.10.5 Equations (7.20) and (7.21)......Page 91
7.10.6 Equations(7.22) and (7.24)......Page 92
References......Page 93
8 Exponent Estimation......Page 94
References......Page 99
9.1 Socioeconomic Perspective......Page 100
9.2 Disparity Curve......Page 101
9.3 Disparity-Curve Analysis......Page 102
9.4 Lorenz Curves......Page 103
9.5 Lorenz-Curves Analysis......Page 105
9.6 Inequality Indices......Page 106
9.7 Gini Index......Page 107
9.8 Reciprocation Index......Page 109
9.9 Summary......Page 110
9.10.1 Equation (9.4)......Page 111
References......Page 112
10.1 Scale Invariance......Page 114
10.2 Perturbation Invariance......Page 115
10.3 Symmetric Perturbations......Page 116
10.4 Socioeconomic Invariance......Page 117
10.5 Poor Fractality and Rich Fractality......Page 119
10.6 Renormalization......Page 120
10.7 Summary......Page 121
10.8.1 Equations(10.1), (10.2), and (10.11)......Page 122
10.8.2 Equations(10.3) and (10.4)......Page 123
10.8.3 Equations(10.5) and (10.6)......Page 124
10.8.4 Equations(10.7) and (10.8)......Page 125
References......Page 126
11.1 One-Sided Lévy Law I......Page 127
11.2 Symmetric Lévy Law I......Page 128
11.3 Uniform Random Scattering......Page 129
11.4 One-Sided Lévy Law II......Page 130
11.5 Symmetric Lévy Law II......Page 131
11.7.1 Equations(11.2) and (11.3)......Page 132
11.7.2 Equations (11.5) and (11.6)......Page 133
References......Page 135
12.1 Growth and Decay......Page 136
12.2 Evolution......Page 137
12.3 Vanishing and Exploding......Page 138
12.4 Order Statistics......Page 139
12.5 Beyond the Singularity......Page 140
12.7 Methods......Page 141
References......Page 142
13.1 Limit Laws I......Page 143
13.2 Limit Laws II......Page 144
13.3 Limit Laws III......Page 146
13.4 Limit Laws IV......Page 147
13.5 Limit Laws V......Page 149
13.6 Outlook......Page 150
13.7.1 Preparation I......Page 153
13.7.3 The Limit mathcalE- via Eq. (13.7)......Page 155
13.7.4 The Limit mathcalE+ via Eq. (13.9)......Page 156
13.7.5 The Limit mathcalE- via Eq. (13.9)......Page 157
13.7.6 Preparation II......Page 158
13.7.7 The Limit mathcalE+ via Eq. (13.19)......Page 159
13.7.8 The Limit mathcalE- via Eq. (13.19)......Page 160
References......Page 161
14 First Digits......Page 162
References......Page 166
15.1 Aggregation......Page 167
15.2 Diffusive Motions......Page 168
15.4 Anomalous Diffusion......Page 169
15.5 Stationary Velocities......Page 171
15.6 White Noise......Page 172
15.7 Flicker Noise......Page 173
15.8 Fusion......Page 174
15.9 Outlook......Page 175
15.10.1 Equation(15.4)......Page 177
15.10.3 Anomalous-Diffusion Examples......Page 178
15.10.4 Equation(15.10)......Page 180
15.10.6 Equation(15.14)......Page 182
References......Page 183
16 First Passage Times......Page 185
References......Page 189
17.1 Double-Pareto Laws......Page 190
17.2 Langevin and Gibbs......Page 192
17.3 Exponentiation......Page 193
17.4 U-Shaped Potentials......Page 194
17.5 Edge of Convexity......Page 196
17.6 Universal Approximation......Page 198
17.7 Lognormal and Log-Laplace Scenarios......Page 199
17.8 Summary......Page 200
17.9.2 Equations(17.14) and (17.15)......Page 201
References......Page 203
18 Conclusion......Page 205
