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

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 $x^{\ast }=0.66692\pm 0.00003$, a value slightly lower than previously estimated yet still distinctly above $2/3$. We also study how the cutoff values $x_N^{\ast }$ at finite $N$ approximate the conjectured value $x^{\ast }$ at $N=\infty$. Assuming $x_N^{\ast }-x_{\infty }^{\ast }\sim N^{-\nu }$, we find that $\nu =0.978\pm 0.025$, which is significantly lower than previous estimates ($\nu \approx 1.4$).

18 figures, 12 pages