[Paper Review] 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 Analysis66 references16 citations

TL;DR

This paper applies branching process theory to model natural hazards like earthquakes, showing that power-law distributions of energy release emerge only at criticality—when the system is poised between extinction and explosion. Using probability generating functions and self-organized criticality, it demonstrates that such systems naturally evolve toward critical states, explaining scale-invariant, heavy-tailed statistics observed in real hazards.

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.

Motivation & Objective

• To explain the origin of power-law distributed energies in natural hazards like earthquakes using branching processes.
• To show that such heavy-tailed statistics arise only at criticality, where the branching process is poised between extinction and explosion.
• To introduce self-organized criticality as a mechanism by which complex systems naturally evolve toward critical states without fine-tuning.
• To provide a self-contained, accessible derivation of key statistical properties using elementary probability and generating functions.
• To draw analogies with statistical physics and emphasize the role of feedback mechanisms in maintaining criticality in natural systems.

Proposed method

• Uses Galton-Watson branching processes to model earthquake aftershock sequences and energy propagation.
• Applies probability generating functions to derive extinction probabilities and size distributions of branching events.
• Derives the critical branching condition where the mean number of offspring equals one, leading to power-law behavior.
• Introduces self-organized criticality (SOC) models with feedback mechanisms that drive the system toward criticality.
• Transforms the Gutenberg-Richter law from magnitude to energy using the relation $ E \propto 10^{3M/2} $, yielding $ D_E(E) \propto E^{-\alpha} $ with $ \alpha = 1 + \frac{2b}{3} $.
• Employs Stirling’s approximation to analyze asymptotic behavior of Catalan numbers and branching process sizes.

Experimental results

Research questions

• RQ1 Why do natural hazard energies follow a power-law distribution with no characteristic scale?
• RQ2 Under what conditions does a branching process produce a power-law distribution of event sizes?
• RQ3 How can a system naturally evolve toward a critical state without external tuning?
• RQ4 What is the role of feedback mechanisms in maintaining criticality in complex systems?
• RQ5 How does the transformation from magnitude to energy distribution affect the observed statistics?

Key findings

• The energy distribution of earthquakes follows a power law $ D_E(E) \propto E^{-\alpha} $ with $ \alpha = 1 + \frac{2b}{3} $, where $ b \approx 1 $, implying $ \alpha \approx \frac{5}{3} $.
• Power-law behavior emerges only at criticality, when the mean number of offspring in a branching process is exactly one.
• For $ p \leq 1/2 $, the total size distribution of branching processes sums to 1, indicating extinction is certain; for $ p > 1/2 $, the sum is $ (q/p)^2 $, indicating a non-zero probability of infinite growth.
• The generating function for Catalan numbers $ h(x) = \frac{1 - \sqrt{1 - 4x}}{2x} $ is used to derive the size distribution of critical branching processes.
• Stirling’s approximation $ n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n $ is derived probabilistically via the gamma distribution and central limit theorem.
• The system exhibits scale invariance and infinite mean energy in theory, making the sample mean an unreliable estimator of the true population mean.