Abstract
We investigate a class of models related to the Bak-Sneppen (BS) model, initially proposed to study evolution. In this model, random fitnesses in [0, 1] are associated with N agents located at the vertices of a graph G, in our case a cycle. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the lowest fitness and its neighbors on the graph G are replaced by new agents with random fitnesses. This simple model after 30+ years still defies exact solution, but captures some forms of complex behavior observed in physical and biological systems.
We use order statistics to define a dynamical system on the set of cumulative distribution functions R : [0,1] → [0,1 that mimics the evolution of the distribution of the fitnesses in these models. We then show that this dynamical system reduces to a 1-dimensional polynomial map. Using an additional conjecture we can then find the limiting distribution as a function of the initial conditions.
Roughly speaking, this ansatz says that the bulk of the replacements in the Bak-Sneppen model occur in a decreasing fraction of the population as the number N of agents tends to infinity. Agreement with experimental results of the BS model is excellent.