Extremal dynamics on complex networks: Analytic solutions

TL;DR

This paper derives an analytical expression for the fitness threshold in the Bak-Sneppen model on complex networks, revealing how network topology influences critical behavior and avalanche size distributions.

Contribution

It provides the first analytic solution for the fitness threshold on complex networks using rate equations and random walk approaches.

Findings

The fitness threshold x_c is 1/(<k>_f+1), depending on degree distribution moments.

Threshold behavior varies with degree exponent γ, being zero or finite.

Avalanche size distribution follows a power law with exponent -3/2.

Abstract

The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold $x_c$ to be 1/(<k>_f+1), where <k>_f=<k^2>/<k> (=<k>) in the quenched (annealed) updating case, where <k^n> is the n-th moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent \gamma <3 (\gamma > 3) for the quenched case in the thermodynamic limit. The theoretical value x_c fits well to the numerical simulation data in the annealed case only. Avalanche size, defined as the duration of successive mutations below the threshold, exhibits a critical behavior as its distribution follows a power law, P_a(s) ~ s^{-3/2}.