Damage Spreading in the Bak–sneppen Model: Sensitivity to the Initial Conditions and Equilibration Dynamics

Abstract

The short-time and long-time dynamics of the Bak-Sneppen model of biological evolution are investigated using the damage spreading technique. By defining a proper Hamming distance measure, we are able to make it exhibit an initial power-law growth which, for finite size systems, is followed by a decay towards equilibrium. In this sense, the dynamics of self-organized critical states is shown to be similar to the one observed at the usual critical point of continuous phase-transitions and at the onset of chaos of non-linear low-dimensional dynamical maps. The transient, pre-asymptotic and asymptotic exponential relaxation of the Hamming distance between two initially uncorrelated equilibrium configurations is also shown to be fit within a single mathematical framework.

Convergence dynamics of 2-dimensional isotropic and anisotropic Bak–Sneppen models

“…After the initial power-law regime ends up at a characteristic time which depends on the system size, Hamming distance reaches a maximum and starts to decrease. The topological behavior of the short-time evolution of the Hamming distance in 2-dimensional BS model is similar to the behavior obtained in 1-dimensional BS model [11]. In order to study the long-time dynamical regime of the Hamming distance evolution one needs to use a slightly different approach than the one mentioned above for the short-time dynamical regime.…”

“…2 it is seen that the pre-asymptotic regime is very close to a slow logarithmic decay which is associated with the uncorrelated nature of local variables in statistically stationary states. Comparing our results of 2-dimensional BS models with the 1-dimensional case [11,13] reveals that the long-time evolution of the minimum Hamming distance measure of both versions of the BS model exhibit topologically similar behavior.…”

“…Therefore, the sensitivity to the initial conditions in the BS model is a crucial matter. In order to study the effects of an initial perturbation one can borrow the damage spreading technique from dynamical systems theory, which has been used in the literature to investigate the propagation of local perturbations in 1-dimensional [6][7][8][9][10][11][12] and 2-dimensional [5] BS models. The algorithm of this technique for a 2-dimensional BS model considered in this work can be described as follows: (i) once the stationary state has been achieved, consider the configuration as replica 1 denoted by f 1 i,j , (ii) produce an identical copy of f 1 i,j and introduce a small damage in this copy by interchanging the site with minimum fitness with a randomly chosen site (denote the new replica as f 2 i,j ), (iii) let both replicas evolve in time using always the same set of random numbers.…”

Introduction to Nonextensive Statistical Mechanics

“…Several studies have been done in connection with self-organized criticality (SOC), in connection with biological evolution [847,849,[851][852][853], imitation games [848], atmospheric cascades [850], earthquakes [855,856], and others [854].…”

Critical behavior of a stochastic anisotropic Bak–Sneppen model

“…The convergence dynamics (both short-time and longtime) of the BS model have been analyzed in Refs. [38][39][40].…”