Rank-driven Markov processes

TL;DR

This paper analyzes a class of rank-driven Markov processes inspired by evolutionary models, establishing threshold behavior and convergence properties for large systems, extending prior results with rigorous proofs and new technical tools.

Contribution

It provides a general theoretical framework for rank-driven Markov processes, extending previous special-case results with rigorous proofs and new convergence techniques.

Findings

System exhibits threshold behavior at a critical value s*

Marginal distribution converges to U[s*, 1] as time and system size grow

Results extend and generalize previous models like Bak--Sneppen

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 $\kappa_N$ on $\{1,2,...,N\}^K$. Our main results are that, under appropriate conditions on $\kappa_N$, the system exhibits threshold behaviour at $s^* \in [0,1]$, where $s^*$ is a function of $\kappa_N$, and the marginal distribution of a randomly selected element converges to $U[s^*, 1]$ as $t \to \infty$ and $N \to \infty$. Of this class of models, results in the literature have previously been given for special cases only, namely the 'mean-field' or 'random neighbour' 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.