Weakly reinforced P\'olya urns on countable networks

TL;DR

This paper analyzes the long-term behavior of weakly reinforced Pólya urns on countable networks, showing positive reinforcement proportions and equilibrium states, especially in regular graphs, for certain reinforcement parameters.

Contribution

It extends the WARM model to countable networks and establishes conditions for positive reinforcement and equilibrium states based on the reinforcement parameter.

Findings

Edges are reinforced positively over time for α<1/2 on bounded degree networks.

In regular graphs, homogenization persists beyond α=1/2.

Reinforcement proportions converge to an equilibrium in the network.

Abstract

We study the long-time asymptotics of a network of weakly reinforced P\'olya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power $\alpha < 1$, and then this weight is increased by $1$. We show that for $\alpha < 1/2$ on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold $\alpha = 1/2$.