论文标题
在可数网络上弱加强的pólyaurns
Weakly reinforced Pólya urns on countable networks
论文作者
论文摘要
我们研究了弱加强Pólyaurn网络的长期渐近学。在此系统中，扩展了R. van der Hofstad等引入的温暖。 al。 （2016年）到可数网络，节点在泊松点过程时有时会发射。当节点启动时，选择了一个事件边缘，其概率与其重量升高到功率$α<1 $的概率成正比，然后该重量增加了$ 1 $。 我们表明，对于$α<1/2 $，在有界度的网络上，每个边缘都得到了积极的时间比例，并且可以将限制比例解释为可计数网络中的平衡。此外，在常规图的特殊情况下，这种均质化仍然有效，超过阈值$α= 1/2 $。
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 $α= 1/2$.
