Graph-based Polya's urn: completion of the linear case

TL;DR

This paper extends the analysis of a graph-based Polya's urn process, proving almost sure convergence of ball proportions for balanced bipartite graphs, where the limit can be a point or an interval, unlike previous non-balanced cases.

Contribution

It establishes almost sure convergence for balanced bipartite graphs and characterizes the possible limit sets, expanding understanding of Polya's urn on graphs.

Findings

Convergence to a point or interval for balanced bipartite graphs.

Extension of convergence results to a new class of graphs.

Characterization of limit sets in the balanced bipartite case.

Abstract

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. Previous works proved that when $G$ is not balanced bipartite, the proportion of balls in the bins converges to a point $w(G)$ almost surely. We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point $w(G)$ or a closed interval $\mathcal J(G)$.