A New Estimate of the Cutoff Value in the Bak-Sneppen Model

TL;DR

This paper revises the estimated cutoff value in the Bak-Sneppen model using extensive simulations, revealing it to be slightly below previous estimates and providing insights into finite-size effects.

Contribution

It introduces a new estimate of the cutoff value and a novel simulation algorithm enabling analysis of larger populations in the Bak-Sneppen model.

Findings

Estimated cutoff value x* = 0.66692 ± 0.00003

Finite-size scaling exponent ν = 0.978 ± 0.025

Model requires N^3 iterations to reach equilibrium

Abstract

We present evidence that the Bak-Sneppen model of evolution on $N$ vertices requires $N^3$ iterates to reach equilibrium. This is substantially more than previous authors suggested (on the order of $N^2$). Based on that estimate, we present a novel algorithm inspired by previous rank-driven analyses of the model allowing for direct simulation of the model with populations of up to $N = 25600$ for $2\cdot N^3$ iterations. These extensive simulations suggest a cutoff value of $x^* = 0.66692 \pm 0.00003$, a value slightly lower than previously estimated yet still distinctly above $2/3$. We also study how the cutoff values $x^*_N$ at finite $N$ approximate the conjectured value $x^*$ at $N=\infty$. Assuming $x^*_N-x^*_\infty \sim N^{-\nu}$, we find that $\nu=0.978\pm 0.025$, which is significantly lower than previous estimates ($\nu\approx 1.4$).