[论文解读] Criticality and self-organization in branching processes: application to natural hazards

Álvaro Corral, Francesc Font-Clos|arXiv (Cornell University)|Jul 11, 2012

Complex Systems and Time Series Analysis参考文献 66被引用 16

一句话总结

本文将分支过程理论应用于地震等自然灾害的建模，表明只有在系统处于临界状态（即灭绝与爆发之间）时，能量释放才会出现幂律分布。通过概率生成函数和自组织临界性理论，证明此类系统自然演化至临界态，从而解释现实中观测到的尺度不变、重尾统计特性。

ABSTRACT

The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that the expected value of the energy does not exist (is infinite), and in practice, that the mean of a finite set of data in not representative of the full population. Also, the distribution presents scale invariance, which implies that it is not possible to define a characteristic scale for the energy. A simple model to account for this peculiar statistics is a branching process: the activation or slip of a fault segment can trigger other segments to slip, with a certain probability, and so on. Although not recognized initially by seismologists, this is a particular case of the stochastic process studied by Galton and Watson one hundred years in advance, in order to model the extinction of (prominent) families. Using the formalism of probability generating functions we will be able to derive, in an accessible way, the main properties of these models. Remarkably, a power-law distribution of energies is only recovered in a very special case, when the branching process is at the onset of attenuation and intensification, i.e., at criticality. In order to account for this fact, we introduce the self-organized critical models, in which, by means of some feedback mechanism, the critical state becomes an attractor in the evolution of such systems. Analogies with statistical physics are drawn. The bulk of the material presented here is self-contained, as only elementary probability and mathematics are needed to start to read.

研究动机与目标

• 通过分支过程解释地震等自然灾害中能量幂律分布的起源。
• 表明此类重尾统计仅在临界状态下出现，即分支过程处于灭绝与爆发之间的平衡点。
• 引入自组织临界性作为复杂系统自然演化至临界态的机制，无需精细调节参数。
• 使用基础概率与生成函数，提供关键统计性质的自包含、易懂推导。
• 与统计物理类比，强调反馈机制在维持自然系统临界性中的作用。

提出的方法

• 使用Galton-Watson分支过程模拟地震余震序列及能量传播。
• 应用概率生成函数推导灭绝概率与分支事件的大小分布。
• 推导临界分支条件（即后代的数学期望等于1），从而导致幂律行为。
• 引入具有反馈机制的自组织临界性（SOC）模型，使系统趋向临界态。
• 通过关系式 $ E \propto 10^{3M/2} $ 将古登堡-里希特律从震级转换为能量，得到 $ D_E(E) \propto E^{-\alpha} $，其中 $ \alpha = 1 + \frac{2b}{3} $。
• 使用Stirling近似分析卡塔兰数与分支过程规模的渐近行为。

实验结果

研究问题

• RQ1 为何自然灾害的能量分布呈现无特征尺度的幂律分布？
• RQ2 在何种条件下分支过程会产生事件规模的幂律分布？
• RQ3 系统如何在无外部调参的情况下自然演化至临界态？
• RQ4 反馈机制在维持复杂系统临界性中起什么作用？
• RQ5 从震级到能量分布的转换如何影响观测到的统计特性？

主要发现

• 地震的能量分布服从幂律 $ D_E(E) \propto E^{-\alpha} $，其中 $ \alpha = 1 + \frac{2b}{3} $，且 $ b \approx 1 $，故 $ \alpha \approx \frac{5}{3} $。
• 幂律行为仅在临界状态下出现，即分支过程的平均后代数恰好为1。
• 当 $ p \leq 1/2 $ 时，分支过程的总规模分布之和为1，表明灭绝必然发生；当 $ p > 1/2 $ 时，总和为 $ (q/p)^2 $，表明存在无限增长的非零概率。
• 利用卡塔兰数的生成函数 $ h(x) = \frac{1 - \sqrt{1 - 4x}}{2x} $ 推导临界分支过程的规模分布。
• 通过伽马分布与中心极限定理，概率性地推导出Stirling近似 $ n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n $。
• 系统表现出尺度不变性，理论上具有无限均值能量，导致样本均值无法可靠估计真实总体均值。