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- W121433478 abstract "We consider the generalized P'{o}lya's Urn model introduced in cite{benjamini2012generalized}. 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 to some fixed power $alpha>0$. 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 $G$ when the reinforcement is linear ($alpha=1$). 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." @default.
- W121433478 created "2016-06-24" @default.
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- W121433478 date "2013-06-23" @default.
- W121433478 modified "2023-09-27" @default.
- W121433478 title "Generalized P'{o}lya's Urn: convergence at linearity" @default.
- W121433478 hasPublicationYear "2013" @default.
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