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

   
@article{ECP3094,
	author = {Jun Chen and Cyrille Lucas},
	title = {A generalized Pólya's urn with graph based interactions: convergence at linearity},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {Dynamical system approach, graph based interactions, ordinary differential equations, Polya's urn, stochastic approximations},
	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.
},
	pages = {no. 67, 1-13},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-3094},    
        url = {http://ecp.ejpecp.org/article/view/3094}}