Bak–Sneppen type models and rank-driven processes

Michael Grinfeld, Philip A. Knight, and Andrew R. Wade

Physical Review E, 84, no. 4, October 2011, article 041124 (5pp). DOI: 10.1103/PhysRevE.84.041124

Abstract

The Bak–Sneppen model is a well-known stochastic model of evolution that exhibits self-organized criticality; only a few analytical results have been established for it so far. 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 Bak–Sneppen model can be replicated by a model with a purely rank-based update rule whose asymptotics can be studied rigorously.