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 (15)

• Avalanche Dynamics in Evolution, Growth, and Depinning Models
• Paths to Self-Organized Criticality
• A Complexity View of Rainfall
• Rain: Relaxations in the sky
• Criticality in Simple Models of Evolution
• On the High-dimensional Bak-Sneppen model
• The Anisotropic Bak-Sneppen Model
• Rain, power laws, and advection
• Critical exponents of the anisotropic Bak-Sneppen model
• d_c=4 is the upper critical dimension for the Bak-Sneppen model
• A perturbative approach to the Bak-Sneppen Model
• Bak-Sneppen model near zero dimension
• Temperature scaling, glassiness and stationarity in the Bak-Sneppen model
• On singular probability densities generated by extremal dynamics
• A Monte Carlo Renormalization Group Approach to the Bak-Sneppen model

Cited by in corpus (6)

• Rank-driven Markov processes
• Bak--Sneppen type models and rank-driven processes
• On Rank Driven Dynamical Systems
• Absorbing-state phase transitions with extremal dynamics
• Critical behavior of a stochastic anisotropic Bak-Sneppen model
• A New Estimate of the Cutoff Value in the Bak-Sneppen Model