On singular probability densities generated by extremal dynamics

TL;DR

This paper investigates the singular probability densities generated by extremal dynamics in the Bak-Sneppen model and its variants, demonstrating their robustness and providing theoretical insights through mean-field analysis and simulations.

Contribution

It introduces a mean-field theoretical framework for understanding singular densities in extremal dynamics models and shows their universality across variants.

Findings

Probability density $p(x)$ is singular at certain points due to extremal dynamics.

The extremal barrier always lies in the 'prohibited' interval where $p(x)=0.

The universality class remains consistent across different updating rules.

Abstract

Extremal dynamics is the mechanism that drives the Bak-Sneppen model into a (self-organized) critical state, marked by a singular stationary probability density $p(x)$. With the aim of understanding this phenomenon, we study the BS model and several variants via mean-field theory and simulation. In all cases, we find that $p(x)$ is singular at one or more points, as a consequence of extremal dynamics. Furthermore we show that the extremal barrier $x_i$ always belongs to the `prohibited' interval, in which $p(x)=0$. Our simulations indicate that the Bak-Sneppen universality class is robust with regard to changes in the updating rule: we find the same value for the exponent $\pi$ for all variants. Mean-field theory, which furnishes an exact description for the model on a complete graph, reproduces the character of the probability distribution found in simulations. For the modified processes mean-field theory takes the form of a functional equation for $p(x)$.