On extinction and survival of the Bak-Sneppen model on arbitrary graphs

TL;DR

This paper extends the analysis of the Bak-Sneppen model's extinction and survival properties from specific cases to a broad class of graphs, using coupling with oriented percolation to simplify the analysis.

Contribution

It generalizes previous results on the Bak-Sneppen model to arbitrary graphs and introduces a coupling method with oriented percolation to streamline proofs.

Findings

Extended non-triviality results to arbitrary graphs

Simplified analysis by coupling with oriented percolation

Established conditions for extinction and survival on various graphs

Abstract

We study the discrete Bak-Sneppen model introduced by Barbay and Kenyon (2001) "On the discrete Bak-Sneppen model of self-organized criticality". We extend their results as well as the non-triviality result of Meester and Znamenskiy (2002) for a finite segment of $\mathbb{Z}^1$ with the periodic boundary condition to a large class of graphs, by using coupling between the Bak-Sneppen model and the oriented percolation in a quadrant. This allows us to avoid dealing with the so-called avalanches, thus simplifying many arguments.