On the thresholds, probability densities, and critical exponents of Bak-Sneppen-like models

Abstract

We report a simple method to accurately determine the threshold and the exponent $\nu$ of the Bak-Sneppen model and also investigate the BS universality class. For the random-neighbor version of the BS model, we find the threshold $x^{\ast }=0.33332(3)$, in agreement with the exact result $x^{\ast }=1/3$ given by mean-field theory. For the one-dimensional original model, we find $x^{\ast }=0.6672(2)$ in good agreement with the results reported in the literature; for the anisotropic BS model we obtain $x^{\ast }=0.7240(1)$. We study the finite size effect $x^{\ast }(L)-x^{\ast }(L\to \infty )\propto L^{-\nu }$, observed in a system with $L$ sites, and find $\nu =1.00(1)$ for the random-neighbor version, $\nu =1.40(1)$ for the original model, and $\nu =1.58(1)$ for the anisotropic case. Finally, we discuss the effect of defining the extremal site as the one which minimizes a general function $f(x)$, instead of simply $f(x)=x$ as in the original updating rule. We emphasize that models with extremal dynamics have singular stationary probability distributions $p(x)$. Our simulations indicate the existence of two symmetry-based universality classes.

8 pages, 3 figures

References in corpus (1)

• On singular probability densities generated by extremal dynamics

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