A generalized Pólya's urn with graph based interactions: convergence at linearity

Cyrille Lucas,Jun Chen

doi:10.1214/ecp.v19-3094

Abstract

We consider a special case of the generalized Pólya's urn model. 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.

Highlights

Introduction and statement of resultsAs a special case of the generalized Pólya’s urn model introduced in [3], the model with linear reinforcement is defined as follows
If G is not balancedbipartite, 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

Summary

Introduction and statement of results

As a special case of the generalized Pólya’s urn model introduced in [3], the model with linear reinforcement is defined as follows. In [3], the authors proved that when G is not balanced-bipartite, the limit of x(n) exists, and it can only take finitely many possible values. Let G be a finite, connected, not balanced-bipartite graph. To prove Theorem 1.1, the main work is to prove the uniqueness of a non-unstable equilibrium for any not balanced-bipartite G. To prove Theorem 1.2, one main difficulty is that the limit set Ω attracts exponentially in the interior, but not at its two endpoints. We will use the limit set theorem stated therein, which requires the following conditions:. [3, Corollary 1.3] Let G be a finite, connected, not balanced-bipartite graph. After proving that the limit set of x(n) is contained in Λ in Proposition 2.2, we want to understand which equilibrium x(n) can converge to. It can be seen from these conditions that only boundary equilibria can be unstable

Not balanced-bipartite graphs

Regular bipartite graphs

Non-regular balanced-bipartite graphs