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On singular probability densities generated by extremal dynamics

Author

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  • Garcia, Guilherme J.M.
  • Dickman, Ronald

Abstract

Extremal dynamics is the mechanism that drives the Bak-Sneppen model into a (self-organized) critical state, marked by a singular stationary probability density p(x). With the aim of understanding this phenomenon, we study the BS model and several variants via mean-field theory and simulation. In all cases, we find that p(x) is singular at one or more points, as a consequence of extremal dynamics. Furthermore we show that the extremal barrier xi always belongs to the ‘prohibited’ interval, in which p(x)=0. Our simulations indicate that the Bak-Sneppen universality class is robust with regard to changes in the updating rule: we find the same value for the exponent π for all variants. Mean-field theory, which furnishes an exact description for the model on a complete graph, reproduces the character of the probability distribution found in simulations. For the modified processes mean-field theory takes the form of a functional equation for p(x).

Suggested Citation

  • Garcia, Guilherme J.M. & Dickman, Ronald, 2004. "On singular probability densities generated by extremal dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 332(C), pages 318-336.
    • Handle: RePEc:eee:phsmap:v:332:y:2004:i:c:p:318-336
    DOI: 10.1016/j.physa.2003.09.047
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