Introducing a real time scale into the Bak-Sneppen model.

Abstract

Recently, a simple model of evolution has been proposed by Bak and Sneppen. This model self-organizes into a critical state for nearest- and random-neighbor interactions. The Bak-Sneppen (BS) model has no explicit time scale, because time steps are always identified with an evolutionary step. Therefore, we introduce at each time step a local stochastical update rule. Hence it is possible to observe time steps in which no species are removed from the system. In the following the durations of time steps in which no further evolution occurs are called interevent intervals. We study a random-neighbor version of the model and derive the steady state distribution of the fitnesses. The distributions are the same for synchronous and asynchronous updating rules and resemble the solutions obtained for the mean field BS model. We give an interpretation of the modified BS model as a neural network with random connections. For a concrete choice of the stochastical updating rule, we derive the distribution of the interevent or interspike intervals. It turns out that for parallel updating we get a power law decay, whereas in the case of random sequential updating the distribution is simply an exponential in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}. N is the system size. All analyzical results are supported by numerical simulations.