Synchronization and functional central limit theorems for interacting reinforced random walks

TL;DR

This paper establishes functional central limit theorems for a class of interacting reinforced random walks, revealing their convergence behavior and synchronization properties, with applications to opinion dynamics in evolving networks.

Contribution

It introduces new functional CLTs for time-inhomogeneous interacting reinforced random walks and analyzes their convergence and synchronization rates, including an application to opinion dynamics.

Findings

Random walks converge to a (possibly random) limit.

Synchronization can occur faster than convergence.

Application to opinion dynamics in evolving networks.

Abstract

We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.