Synchronization via interacting reinforcement

TL;DR

This paper studies a system of interacting Polya urns where reinforcement depends on individual urns and the collective average, demonstrating almost sure synchronization of color proportions and providing a normal approximation for large systems.

Contribution

It introduces a model of interacting urns with combined local and global reinforcement, proving synchronization and deriving a normal approximation for large numbers of urns.

Findings

Urns synchronize almost surely in their color proportions.

The fraction of balls converges to a common limit for all urns.

A normal approximation is derived for large urn systems.

Abstract

We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.