Generalizations of the Discrete Bak-Sneppen Model

Sammanfattning: Consider n vertices arranged in a circle where each vertex is given a fitness from {0,1}. At each discrete time step, one of the vertices with fitness equal to zero (unless there are none of those, then pick a vertex with fitness equal to one) is selected with equal probability. Then this vertex and the two neighbouring vertices are each given a new fitness from a Bernoulli(p) distribution independently of each other, for some p in [0,1]. This model is known as the discrete Bak-Sneppen model. What happens to the fraction of ones (vertices with fitness one) as n and the time t goes to infinity? How does this quantity depend on p? Is there a pc in (0,1) such that this quantity is equal to one for p > pc and less than one for p < pc? In this paper we prove upper bounds for pc for generalized versions of this model. We alsoprovide a number of experimental results, as well as a quick summary of what hasbeen done in the past.