Expansion Around the Mean-Field Solution of the Bak-Sneppen Model

TL;DR

This paper develops an analytical and numerical expansion around the mean-field solution for the Bak-Sneppen model, relating avalanche size distribution exponents to fractal dimensions, and validates results with simulations.

Contribution

It introduces an expansion method around the mean-field solution to connect avalanche exponents with fractal dimensions in the Bak-Sneppen model.

Findings

Analytical relation between avalanche exponent and fractal dimension.

Numerical computation of the relation up to second order expansion.

Results agree with Monte Carlo simulations in 1D and 2D.

Abstract

We study a recently proposed equation for the avalanche distribution in the Bak-Sneppen model. We demonstrate that this equation indirectly relates $\tau$,the exponent for the power law distribution of avalanche sizes, to $D$, the fractal dimension of an avalanche cluster.We compute this relation numerically and approximate it analytically up to the second order of expansion around the mean field exponents. Our results are consistent with Monte Carlo simulations of Bak-Sneppen model in one and two dimensions.