Interacting Urns on a Finite Directed Graph
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 outneighbours according to a fixed, nonnegative 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 Pólya type or if the graph is such that every vertex with nonzero indegree can be reached from some vertex with zero indegree. 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 indegree and the reinforcement is of Pólya 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.10738
 Bibcode:
 2019arXiv190510738K
 Keywords:

 Mathematics  Probability