Weakly reinforced Pólya urns on countable networks

Yannick Couzinié,Christian Hirsch

doi:10.1214/21-ecp404

Yannick Couzinié, Christian Hirsch

Open Access

https://doi.org/10.1214/21-ecp404

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Abstract

We study the long-time asymptotics of a network of weakly reinforced Pólya 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 α<1, and then this weight is increased by 1. We show that for α<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 α=1∕2.

Highlights

Pólya’s urn process is the paradigm model for a random process incorporating reinforcement effects
We show that for α < 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
In the field of social sciences, the formation of friendship networks could be related to reinforcement effects in social interactions [7]

Summary

Introduction

Pólya’s urn process is the paradigm model for a random process incorporating reinforcement effects. Together with the analysis in [5, 6], our main result is a first step towards a network-based analog of Rubin’s dichotomy for classical Pólya urns: while for strong reinforcement some edges are only reinforced finitely often, in the weakly reinforced regime all edges are reinforced a positive proportion of time. This description outlines the broader picture, more research is needed to carve out the precise conditions for this dichotomy.

Model definition and main result

Existence of equilibrium

Regular graphs