A generalized Polya's urn with graph based interactions

TL;DR

This paper studies a generalized Polya's urn model on graphs, analyzing how the proportion of balls in bins evolves and converges, revealing conditions for unique or uniform limiting distributions based on graph structure and parameters.

Contribution

It introduces a dynamical systems approach to characterize the limiting behavior of a graph-based Polya's urn with power-dependent probabilities, extending previous models.

Findings

For a<1, the proportions converge to a unique nonzero vector.

In regular, non-bipartite graphs with a≤1, the proportions converge to the uniform distribution.

The limit set is contained within the equilibria of an associated vector field.

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 raised by some fixed power a>0. We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if a<1 then there is a single point v=v(G,a) with nonzero entries such that the proportion converges to v almost surely. A special case is when G is regular and a is at most 1. We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.