Interacting vertex reinforced random walks on complete sub-graphs

This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete subgraph 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 provides a general framework that yields a detailed description of the limiting behaviour for any choice of interaction parameters and subgraph geometry. We illustrate this by analyzing interacting walks on complete graphs, stars, and cycles, chosen to highlight the model's broad applicability. The central contribution lies in offering a powerful tool to analyse diverse types of interactions mediated by the intersections of subgraphs. Importantly, our results provide not only convergence criteria, but also conditions under which the empirical proportions of the walks' visits fail to converge to certain boundary points of the space where their trajectory evolves.