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Article
On Rank Driven Dynamical Systems
Mathematics and Statistics Faculty Publications and Presentations
  • J. J. P. Veerman, Portland State University
  • F. J. Prieto, University of Carlos III, Madrid
Document Type
Post-Print
Publication Date
8-1-2014
Subjects
  • Dynamical Systems,
  • Order statistics
Abstract

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 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 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.

Description

This is the post-print version of an article which was subsequently published in the Journal of Statistical Physics, Copyright (2014) SpringerLink.

Version of record can be found at: http://dx.doi.org/10.1007/s10955-014-1012-0

Erratum is located below in Additional Files

DOI
10.1007/s10955-014-1012-0
Persistent Identifier
http://archives.pdx.edu/ds/psu/17532
Citation Information
J. J. P. Veerman and F. J. Prieto. "On Rank Driven Dynamical Systems" Vol. 156 Iss. 3 (2014) p. 455 - 472
Available at: http://works.bepress.com/jj-veerman/7/