On a Local Version of the Bak–Sneppen 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. Over the years a number of models inspired by Bak–Sneppen were studied, usually by introducing different or new features (e.g. discretizing fitness, randomized neighbors or population size). We present a variant that only uses features present in Bak–Sneppen, and whose difference from the Bak–Sneppen is that only the graph geometry is used for the evolution. This allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model. 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.