On rank driven dynamical systems Articles uri icon

publication date

  • August 2014

start page

  • 455

end page

  • 472

issue

  • 3

volume

  • 156

international standard serial number (ISSN)

  • 0022-4715

electronic international standard serial number (EISSN)

  • 1572-9613

abstract

  • We investigate a class of models related to the Bak-Sneppen (BS) 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 are associated to agents located at the vertices of a graph . Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the worst fitness and some others with a priori given rank probabilities are replaced by new agents with random fitnesses. We consider two cases: The exogenous case where the new fitnesses are taken from an a priori fixed distribution, and the 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 rank-driven dynamical system that approximates the evolution of the distribution of the fitnesses in these rank-driven models, as well as in the BS 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.

keywords

  • dynamical systems; order statistics; asymptotic approximations; bak-sneppen model; biological evolution; criticality; thresholds