Graph-based Pólya’s urn: Completion of the linear case

Abstract

Given a finite connected graph [Formula: see text], place a bin at each vertex. Two bins are called a pair if they share an edge of [Formula: see text]. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. This model was introduced by Benaim, Benjamini, Chen, and Lima. When [Formula: see text] is not balanced bipartite, Chen and Lucas proved that the proportion of balls in the bins converges to a point [Formula: see text] almost surely. We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point [Formula: see text] or a closed interval [Formula: see text].