Weibull-type limiting distribution for replicative systems

TL;DR

This paper demonstrates that the distribution of simple replicative systems modeled by Galton-Watson processes closely resembles the Weibull distribution across various parameters, revealing a universal pattern in natural skew distributions.

Contribution

It establishes a connection between Galton-Watson branching processes and Weibull distributions, providing a new understanding of their ubiquity in natural phenomena.

Findings

Distribution from branching process matches Weibull form

Universal series expansion for cumulative distribution

Mapping of branching to cluster aggregation process

Abstract

The Weibull function is widely used to describe skew distributions observed in nature. However, the origin of this ubiquity is not always obvious to explain. In the present paper, we consider the well-known Galton-Watson branching process describing simple replicative systems. The shape of the resulting distribution, about which little has been known, is found essentially indistinguishable from the Weibull form in a wide range of the branching parameter; this can be seen from the exact series expansion for the cumulative distribution, which takes a universal form. We also find that the branching process can be mapped into a process of aggregation of clusters. In the branching and aggregation process, the number of events considered for branching and aggregation grows cumulatively in time, whereas, for the binomial distribution, an independent event occurs at each time with a given success probability.