Bak-Sneppen model: Local equilibrium and critical value.

The Bak-Sneppen (BS) model is a very simple model that exhibits all the richness of self-organized criticality theory. At the thermodynamic limit, the BS model converges to a situation where all particles have a fitness that is uniformly distributed between a critical value p_{c} and 1. The p_{c} value is unknown, as are the variables that influence and determine this value. Here we study the BS model in the case in which the lowest fitness particle interacts with an arbitrary even number of m nearest neighbors. We show that p_{c} verifies a simple local equilibrium relation. Based on this relation, we can determine bounds for p_{c} of the BS model and exact results for some BS-like models. Finally, we show how transformations of the original BS model can be done without altering the model's complex dynamics.