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


 
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1. Title Title of document A generalized Pólya's urn with graph based interactions: convergence at linearity
 
2. Creator Author's name, affiliation, country Jun Chen; Caltech; United States
 
2. Creator Author's name, affiliation, country Cyrille Lucas; Weizmann Institute of Science and Université Paris Diderot; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Dynamical system approach, graph based interactions, ordinary differential equations, Polya's urn, stochastic approximations
 
3. Subject Subject classification 60K35
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Weizmann Institute of Science, ISF
 
7. Date (YYYY-MM-DD) 2014-10-03
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3094
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3094
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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