On Rank Driven Dynamical Systems

We investigate a class of models related to the Bak-Sneppen model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of "complex behavior" such as self-organized criticality that is often observed in physical and biological systems. In this model, random fitnesses in $[0,1]$ are associated to agents located at the vertices of a graph $G$. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the worst fitness and some others \emph{with a priori given rank probabilities} are replaced by new agents with random fitnesses. We consider two cases: The \emph{exogenous case} where the new fitnesses are taken from an a priori fixed distribution, and the \emph{endogenous case} where the new fitnesses are taken from the current distribution as it evolves. We approximate the dynamics by making a simplifying independence assumption. We use Order Statistics and Dynamical Systems to define a \emph{rank-driven dynamical system} that approximates the evolution of the \emph{distribution} of the fitnesses in these rank-driven models, as well as in the Bak-Sneppen model. For this simplified model we can find the limiting marginal distribution as a function of the initial conditions. Agreement with experimental results of the BS model is excellent.