Critical and Near-Critical Branching Processes

TL;DR

This paper investigates the conditions under which branching processes produce scale-free power-law distributions, analyzing their robustness and applying findings to biological, artificial, and physical systems.

Contribution

It introduces a detailed analysis of critical and near-critical branching processes and their role in generating scale-free dynamics across different systems.

Findings

Identifies conditions for power-law emergence in branching processes

Analyzes how deviations from criticality affect scale-free behavior

Predicts distribution patterns in biological and physical systems

Abstract

Scale-free dynamics in physical and biological systems can arise from a variety of causes. Here, we explore a branching process which leads to such dynamics. We find conditions for the appearance of power laws and study quantitatively what happens to these power laws when such conditions are violated. From a branching process model, we predict the behavior of two systems which seem to exhibit near scale-free behavior--rank-frequency distributions of number of subtaxa in biology, and abundance distributions of genotypes in an artificial life system. In the light of these, we discuss distributions of avalanche sizes in the Bak-Tang-Wiesenfeld sandpile model.