Dynamics of vertex-reinforced random walks

TL;DR

This paper studies the long-term behavior of vertex-reinforced random walks on graphs, showing they tend to localize on specific subgraphs related to stable equilibria of associated replicator dynamics.

Contribution

It generalizes previous results by linking VRRW localization to stable equilibria of replicator dynamics on graphs, applicable to population genetics and game theory.

Findings

VRRW localizes on complete d-partite subgraphs with positive probability.

Stable equilibria support the walk's localization and convergence of occupation density.

The results connect VRRW behavior with replicator dynamics stability analysis.

Abstract

We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena\"{{\i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $S\cup\partial S$ (where $\partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.