可数网络上的弱加强Pólya瓮
Weakly reinforced Pólya urns on countable networks
摘要：
我们研究了一个弱加强的Pólya瓮网络的长时间渐近性。该系统扩展了R. van der Hofstad et. al.（2016）到可数网络，节点在泊松点过程给定的时间触发。当节点激发时，以与其权重的幂$α< 1$成比例的概率选择入射边之一，然后将该权重增加$1$。我们表明，$α< 1/2$在网络上的有界度，每一个边缘加强一个积极的时间比例，并限制比例可以解释为一个可数网络的平衡。此外，在正则图的特殊情况下，这种均匀化在阈值$α= 1/2$之外仍然有效。
英文摘要：
We study the long-time asymptotics of a network of weakly reinforced Pólya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power $α< 1$, and then this weight is increased by $1$. We show that for $α< 1/2$ on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold $α= 1/2$.
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