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

We report a simple method to accurately determine the threshold and the exponent ν of the Bak–Sneppen (BS) model and also investigate the BS universality class. For the random-neighbor version of the BS model, we find the threshold x∗=0.33332(3), in agreement with the exact result x∗=13 given by mean-field theory. For the one-dimensional original model, we find x∗=0.6672(2) in good agreement with the results reported in the literature; for the anisotropic BS model we obtain x∗=0.7240(1). We study the finite size effect x∗(L)−x∗(L→∞)∝L−ν, observed in a system with L sites, and find ν=1.00(1) for the random-neighbor version, ν=1.40(1) for the original model, and ν=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.