Translator Disclaimer
2014 A generalized Pólya's urn with graph based interactions: convergence at linearity
Jun Chen, Cyrille Lucas
Author Affiliations +
Electron. Commun. Probab. 19: 1-13 (2014). 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.

Citation

Download Citation

Jun Chen. Cyrille Lucas. "A generalized Pólya's urn with graph based interactions: convergence at linearity." Electron. Commun. Probab. 19 1 - 13, 2014. https://doi.org/10.1214/ECP.v19-3094

Information

Accepted: 3 October 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1326.60135
MathSciNet: MR3269167
Digital Object Identifier: 10.1214/ECP.v19-3094

Subjects:
Primary: 60K35

JOURNAL ARTICLE
13 PAGES


SHARE
Back to Top