Dynamic of vertwx-reinforced random walks.
Dynamic of vertwx-reinforced random walks.
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vertwx 强化随机游走的动态。
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
P. Tarres
中科院分区：
文献类型：
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作者：
M. Benaim;P. Tarres
我们概括了沃尔科夫 [Ann.很可能。 29 (2001) 66--91] 并证明，在一大类有界度 $(G,sim)$ 和对称强化矩阵 $a=(a_{i,j})_{i,jin G}$ 的局部有限连通图上，顶点强化随机游走 (VRRW) 最终以正概率定位在由完整的 $d$ 部分子图（具有可能的循环及其外部边界）组成的子集上。我们首先证明，一般来说，在图 $G$ 上具有正收益的线性对称复制动态的任何稳定均衡都满足以下性质：对于某些 $dge1$，其支持是 $G$ 的完整 $d$ 部分子图，具有可能的循环。该结果用于 VRRW 的研究，但也适用于其他环境，例如群体遗传学和博弈论中的进化模型。接下来我们概括 Pemantle [概率。理论相关领域 92 (1992) 117--136] 和 Bena"{{i}}m [Ann. Probab. 25 (1997) 361--392] 将 VRRW 的渐近行为与复制器动力学联系起来。这使我们能够得出这样的结论：给定任何具有 $S$ 支撑的严格稳定均衡的邻域，以下事件以正概率发生：行走定位在 $Scuppartial S$ 上（其中 $partial S$ 是 $S$ 的外边界）并且 VRRW 的占用密度以多项式率收敛到该邻域中的严格稳定平衡。
We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,jin G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $dge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena"{{i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $Scuppartial S$ (where $partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.
影响因子：
1
作者：
Cohen N
通讯作者：
Cohen N