Interacting vertex reinforced random walks on complete sub-graphs

TL;DR

This paper introduces a model for interacting vertex-reinforced random walks on complete sub-graphs, analyzing their convergence properties and the effects of interaction parameters on their limiting behaviour.

Contribution

It develops a comprehensive framework for analyzing the almost sure convergence of interacting reinforced walks and explores diverse interaction scenarios and geometries.

Findings

Almost sure convergence to fixed points for most parameters

Limiting behaviour depends on interaction parameters and graph structure

Examples include cooperation and competition dynamics in various sub-graphs

Abstract

This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete sub-graph of a locally finite undirected graph. The transition probability for a walk to a given vertex depends on the cumulative proportion of visits by all walks that have access to that vertex. Proportions are modified by multiplication by a real valued interaction parameter and the addition of a parameter representing the intrinsic preference of the walk for the vertex. This model covers a wide range of interactions, including the cooperation (attraction) or competition (repulsion) of several walks at single vertices. We are principally concerned with strong laws for the proportion of visits to each vertex by all walks. We prove that this measure converges almost surely towards the set of fixed points of the transition probabilities. Almost sure convergence to a single fixed point is in fact the generic behaviour as we show this to hold for almost all parameter values of our model. Beyond almost sure convergence, our model allows for a detailed description of the limiting behaviour depending on the interaction parameters and the sub-graph geometries. This is illustrated by several examples of the competitive version of the dynamics, including interacting walks sharing all the vertices of finite complete graphs and walks confined to complete sub-graphs of star graphs and cycles.