Rank-driven Markov processes

Grinfeld, Michael; Knight, Philip A.; Wade, Andrew R.

Abstract

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κN on {1,2,…,N}K. Our main results are that, under appropriate conditions on κN, the system exhibits threshold behavior at s∗∈[0,1], where s∗ is a function of κN, and the marginal distribution of a randomly selected element converges to U[s∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.

Citation

Grinfeld, M., Knight, P. A., & Wade, A. R. (2012). Rank-driven Markov processes. Journal of Statistical Physics, 146(2), 378-407. https://doi.org/10.1007/s10955-011-0368-7

• Journal Article Type: Article
• Publication Date: Jan 1, 2012
• Deposit Date: Oct 4, 2012
• Publicly Available Date: Jan 31, 2013
• Journal: Journal of Statistical Physics
• Print ISSN: 0022-4715
• Electronic ISSN: 1572-9613
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 146
• Issue: 2
• Pages: 378-407
• DOI: https://doi.org/10.1007/s10955-011-0368-7
• Keywords: Bak–Sneppen evolution model, Self-organized criticality, Markov process on order statistics, Phase transition, Interacting particle system
• Public URL: https://durham-repository.worktribe.com/output/1502776