A generalized P\'olya's Urn with graph based interactions: convergence at linearity

TL;DR

This paper studies a generalized Pólya's urn model on graphs, proving almost sure convergence of ball proportions to deterministic points for non-balanced bipartite graphs and to an interval for regular bipartite graphs, with some cases still open.

Contribution

It extends Pólya's urn analysis to graph-based interactions, providing convergence results for various graph structures and highlighting open problems.

Findings

Proves convergence to a deterministic point for non-balanced bipartite graphs.

Shows convergence to an interval for regular bipartite graphs.

Identifies open problems for non-regular balanced bipartite graphs.

Abstract

We consider a special case of the generalized P\'{o}lya's urn model introduced by Benaim et al (2013). Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. 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. A question of essential interest for the model is to understand the limiting behavior of the proportion of balls in the bins for different graphs $G$. In this paper, we present two results regarding this question. If $G$ is not balanced-bipartite, we prove that the proportion of balls converges to some deterministic point $v=v(G)$ almost surely. If $G$ is regular bipartite, we prove that the proportion of balls converges to a point in some explicit interval almost surely. The question of convergence remains open in the case when $G$ is non-regular balanced-bipartite (see a final remark in the paper).