Phase transition in random adaptive walks on correlated fitness landscapes

TL;DR

This paper analyzes how the length of adaptive walks in correlated fitness landscapes depends on the strength of the fitness gradient, revealing a phase transition for exponential tail distributions.

Contribution

It provides an analytical study of adaptive walk lengths on correlated fitness landscapes, identifying a phase transition in walk length behavior based on the fitness gradient strength.

Findings

For exponential tail distributions, a phase transition occurs at a critical gradient c.

Walk length scales as ln L at small c and as L at large c for exponential distributions.

For other distributions, only a single phase with walk length scaling as ln L exists.

Abstract

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength $c$. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size $L$. When the distribution of the random fitness component has an exponential tail we find a phase transition of the walk length $D$ between a phase at small $c$ where walks are short $(D \sim \ln L)$ and a phase at large $c$ where walks are long $(D \sim L)$. For all other distributions only a single phase exists for any $c > 0$. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.