On a Local Version of the Bak-Sneppen Model

TL;DR

This paper introduces a local variant of the Bak-Sneppen model that simplifies analysis by using only graph geometry, revealing convergence properties and ergodic behavior in large constant-degree graphs.

Contribution

It proposes a new local version of the Bak-Sneppen model that enables stationary distribution analysis via random walk dynamics, differing from traditional global approaches.

Findings

Stationary fitness distribution converges to IID law in large graphs

Model exhibits exponential ergodicity through coupling

Analysis of avalanches in the local model

Abstract

A major difficulty in studying the Bak-Sneppen model is in effectively comparing it with well-understood models. This stems from the use of two geometries: complete graph geometry to locate the global fitness minimizer, and graph geometry to replace the species in the neighborhood of the minimizer. We present a variant in which only the graph geometry is used. This allows to obtain the stationary distribution through random walk dynamics. We use this to show that for constant-degree graphs, the stationary fitness distribution converges to an IID law as the number of vertices tends to infinity. We also discuss exponential ergodicity through coupling, and avalanches for the model.