Branching Process in a Stochastic Extremal Model

TL;DR

This paper introduces a stochastic variant of the Bak-Sneppen model with limited mutation sites, analyzing its critical behavior across different graph structures and demonstrating its self-organized criticality under various conditions.

Contribution

It proposes a new stochastic model with restricted mutation zones, explores its critical behavior on different graphs, and conjectures its criticality on arbitrary graphs with small branching factors.

Findings

Critical behavior matches the Bak-Sneppen model in 1D and 2D.

On scale-free graphs, the critical fitness value remains non-zero in the thermodynamic limit.

Reducing the mutation zone size still reproduces Bak-Sneppen behavior.

Abstract

We considered a stochastic version of the Bak-Sneppen model (SBSM) of ecological evolution where the the number $M$ of sites mutated in a mutation event is restricted to only two. Here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion. The critical behavior of the SBSM is found to be the same as the BS model in dimensions $d$ =1 and 2. However on the scale-free graphs the critical fitness value is non-zero even in the thermodynamic limit but the critical behavior is mean-field like. Finally $<M>$ has been made even smaller than two by probabilistically updating the mutation zone which also shows the original BS model behavior. We conjecture that a SBSM on any arbitrary graph with any small branching factor greater than unity will lead to a self-organized critical state.