Interacting Urns on a Finite Directed Graph

• Main Authors: Kaur, Gursharn, Sahasrabudhe, Neeraja
• Format: Journal Article
• Language: English
• Published: 26-05-2019
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.1905.10738

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.