arXiv 2010.095962020-10math.PR

有向随机图上的随机递归

Stochastic recursions on directed random graphs

Nicolas Fraiman, Tzu-Chi Lin, Mariana Olvera-Cravioto

摘要：

对于顶点$V_n = \{1,2, \dots, n\}$上的有向图$G(V_n, E_n)$，我们研究了$\mathbb{R}^n$上的Markov链$\{ {\bf R}^{(k)}: k \geq 0\}$的分布，使得${\bf R}^{(k)}$的$i$第n个分量，记为$R_i^{(k)}$，对应于顶点$i$上的过程在时刻$k$的值.我们专注于进程$\{ {\bf R}^{(k)}: k \geq 0\}$的值$R_i^{(k+1)}$只取决于它的入站邻居的值$\{ R_j^{(k)}: j \to i\}$，并可能在顶点属性。然后，我们表明，提供$G(V_n, E_n)$收敛在局部弱意义上的一个标记的高尔顿-沃森过程，动态的过程中，一个统一选择的顶点$V_n$可以耦合，对于任何固定的$k$，一个过程$\{ \mathcal{R}_\emptyset^{(r)}: 0 \leq r \leq k\}$上构建的限制标记的高尔顿-沃森树。此外，我们推导出充分条件下，$\mathcal{R}^{(k)}_\emptyset$收敛，$k \to \infty$，一个随机变量$\mathcal{R}^*$，其特征在于吸引内生解的分支分布不动点方程。我们的框架也可以应用到进程$\{ {\bf R}^{(k)}: k \geq 0\}$，其唯一的随机性来源于图$G(V_n, E_n)$的实现。

英文摘要：

For a directed graph $G(V_n, E_n)$ on the vertices $V_n = \{1,2, \dots, n\}$, we study the distribution of a Markov chain $\{ {\bf R}^{(k)}: k \geq 0\}$ on $\mathbb{R}^n$ such that the $i$th component of ${\bf R}^{(k)}$, denoted $R_i^{(k)}$, corresponds to the value of the process on vertex $i$ at time $k$. We focus on processes $\{ {\bf R}^{(k)}: k \geq 0\}$ where the value of $R_i^{(k+1)}$ depends only on the values $\{ R_j^{(k)}: j \to i\}$ of its inbound neighbors, and possibly on vertex attributes. We then show that, provided $G(V_n, E_n)$ converges in the local weak sense to a marked Galton-Watson process, the dynamics of the process for a uniformly chosen vertex in $V_n$ can be coupled, for any fixed $k$, to a process $\{ \mathcal{R}_\emptyset^{(r)}: 0 \leq r \leq k\}$ constructed on the limiting marked Galton-Watson tree. Moreover, we derive sufficient conditions under which $\mathcal{R}^{(k)}_\emptyset$ converges, as $k \to \infty$, to a random variable $\mathcal{R}^*$ that can be characterized in terms of the attracting endogenous solution to a branching distributional fixed-point equation. Our framework can also be applied to processes $\{ {\bf R}^{(k)}: k \geq 0\}$ whose only source of randomness comes from the realization of the graph $G(V_n, E_n)$.

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摘要：

英文摘要：

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