# Interacting Urns on a Finite Directed Graph

## TL;DR

This paper studies a two-colour interacting urn model on a finite directed graph, proving convergence of colour proportions and establishing central limit theorems, with special analysis for Pólya-type reinforcement on regular graphs.

## Contribution

It introduces a general interacting urn model on directed graphs, analyzing convergence and fluctuations, including new results for Pólya-type reinforcement on regular graphs.

## Key findings

- Colour proportions converge almost surely to deterministic or random limits.
- Joint central limit theorems describe fluctuations around the limits.
- Special case analysis for Pólya reinforcement on regular graphs shows uniform convergence to a random limit.

## Abstract

We introduce a general two colour interacting urn model on a finite directed graph, where each urn at a node, reinforces all the urns in its out-neighbours according to a fixed, non-negative and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of P\'olya type or if the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems, with appropriate scaling, around the vector of limiting proportion. Further, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of P\'olya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.