Branching Random Walks on Binary Strings for Evolutionary Processes

TL;DR

This paper analyzes branching random walks on graphs modeling evolutionary processes, revealing that higher mutation rates beyond a threshold do not improve exploration speed, using expander graph theory to estimate cover times.

Contribution

It introduces a novel application of expander graph theory to model and analyze mutation processes in evolutionary branching random walks.

Findings

Higher mutation rates do not increase exploration speed beyond a threshold.

Estimates for cover times of branching random walks are provided.

A saturation phenomenon in mutation rates is identified.

Abstract

In this article, we study branching random walks on graphs modeling division-mutation processes inspired by adaptive immunity. We apply the theory of expander graphs on mutation rules in evolutionary processes and obtain estimates for the cover times of the branching random walks. This analysis reveals an unexpected saturation phenomenon : increasing the mutation rate above a certain threshold does not enhance the speed of state-space exploration.