Bak--Sneppen type models and rank-driven processes

TL;DR

This paper reveals a surprising connection between Bak-Sneppen models and simpler rank-driven Markov processes, enabling rigorous analysis of their long-term behavior without relying on complex topologies.

Contribution

It introduces a novel rank-based model that replicates the long-term dynamics of Bak-Sneppen models, simplifying their analytical study.

Findings

Long-term behavior of Bak-Sneppen models can be captured by rank-driven processes.

Rank-based models allow rigorous asymptotic analysis.

Topology-independent models replicate key features of original models.

Abstract

The Bak--Sneppen model is a simple stochastic model of evolution that exhibits self-organized criticality and for which few analytical results have been established. In the original Bak-Sneppen model and many subsequent variants, interactions among the evolving species are tied to a specified topology. We report a surprising connection between Bak-Sneppen type models and more tractable Markov processes that evolve without any reference to an underlying topology. Specifically, we show that in the case of a large number of species, the long time behaviour of the fitness profile in the anisotropic Bak--Sneppen model can be replicated by a model with a purely rank-based update rule whose asymptotics can be studied rigorously.