dc = 4 is the upper critical dimension for the Bak-Sneppen model

Fitness fluctuations in the Bak-Sneppen model

We study the one-dimensional Bak-Sneppen model for the evolution of species in an ecosystem. Of particular interest are the temporal fluctuations in fitness variables. We numerically compute the power spectral density and apply the finite-size scaling method to get data collapse. A clear signature of 1/f^{α} noise with α≈1.2 (long-time correlations) emerges for both local and global (or average) fitness noises. The limiting value of the spectral exponent, 0 or 2, corresponds to no interaction or a random neighbor version of the model, respectively. The local power spectra are spatially uncorrelated and also show an additional scaling, ∼1/L, in the frequency regime L^{-λ}≪f≪1/2, where L is the linear extent of the system.

Branching process in a stochastic extremal model

We considered a stochastic version of the Bak-Sneppen model (SBSM) of ecological evolution where the number M of sites mutated in a mutation event is restricted to only two. Here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion. The critical behavior of the SBSM is found to be the same as the BS model in dimensions d=1 and 2. However on the scale-free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean-field like. Finally M has been made even smaller than two by probabilistically updating the mutation zone, which also shows the original BS model behavior. We conjecture that a SBSM on any arbitrary graph with any small branching factor greater than unity will lead to a self-organized critical state.

Absorbing-state phase transitions with extremal dynamics

Extremal dynamics represents a path to self-organized criticality in which the order parameter is tuned to a value of zero. The order parameter is associated with a phase transition to an absorbing state. Given a process that exhibits a phase transition to an absorbing state, we define an "extremal absorbing" process, providing the link to the associated extremal (nonabsorbing) process. Stationary properties of the latter correspond to those at the absorbing-state phase transition in the former. Studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible. In the case of the Bak-Sneppen (BS) model, the absorbing version is closely related to the "f -avalanche" introduced by Paczuski, Maslov, and Bak [Phys. Rev. E 53, 414 (1996)], or, in spreading simulations to the "BS branching process" also studied by these authors. The corresponding nonextremal process belongs to the directed percolation universality class. We revisit the absorbing BS model, obtaining refined estimates for the threshold and critical exponents in one dimension. We also study an extremal version of the usual contact process, using mean-field theory and simulation. The extremal condition slows the spread of activity and modifies the critical behavior radically, defining an "extremal directed percolation" universality class of absorbing-state phase transitions. Asymmetric updating is a relevant perturbation for this class, even though it is irrelevant for the corresponding nonextremal class.

Universal persistence exponents in an extremally driven system

The local persistence R(t), defined as the proportion of the system still in its initial state at time t, is measured for the Bak-Sneppen model. For one and two dimensions, it is found that the decay of R(t) depends on one of two classes of initial configuration. For a subcritical initial state, R(t) equivalent to t(-theta), where the persistence exponent theta can be expressed in terms of a known universal exponent. Hence theta is universal. Conversely, starting from a supercritical state, R(t) decays by the anomalous form 1-R(t) equivalent to t(tau(all)) until a finite time t(0), where tau(all) is also a known exponent. Finally, for the high dimensional model R(t) decays exponentially with a nonuniversal decay constant.

Bak-sneppen model near zero dimension

We consider the Bak-Sneppen model near zero dimension, where the avalanche exponent tau is close to 1 and the exponents &mgr; and sigma are close to 0. We demonstrate that tau-1=&mgr;-sigma=exp-&mgr;(-1)-gamma+ small middle dot small middle dot small middle dot in this limit, where gamma is Euler's constant. The avalanche hierarchy equation is rewritten in a form that makes it possible to find the relation between the critical exponents sigma and &mgr; with high accuracy. We obtain precise values of the critical exponents for the one and two-dimensional Bak-Sneppen model and for the one-dimensional anisotropic Bak-Sneppen model.