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.
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机译：我们考虑广义Pólya骨灰盒模型的特殊情况。给定一个有限的连通图$ G $，在每个顶点放置一个bin。如果两个垃圾箱共享$ G $的边缘，则称为一对。在不连续的时间，将球添加到每对箱中。在一对垃圾箱中，一个垃圾箱以与当前球数成正比的概率获得球。模型的主要兴趣问题是了解不同图形在垃圾箱中球比例的限制行为。 G $。在本文中，我们提出了有关该问题的两个结果。如果$ G $不是二等分，我们证明球的比例几乎可以肯定地收敛到某个确定点$ v = v（G）$。如果$ G $是规则的二等分，我们证明球的比例几乎可以肯定地收敛到某个明确间隔内的某个点。在$ G $是非规则的平衡二分的情况下，收敛问题仍然存在。
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