论文标题
定向随机图上的随机递归
Stochastic recursions on directed random graphs
论文作者
论文摘要
对于在顶点上的定向图$ g（v_n，e_n）$，我们研究了马尔可夫链$ \ {{\ bf r^}^{（k）的分布$ {\ bf r}^{（k）} $，表示为$ r_i^{（k）} $，对应于time $ k $在顶点$ i $上的过程的值。我们专注于$ \ {{\ bf r}^{（k）}：k \ geq 0 \} $，其中$ r_i^{（k+1）} $的值仅取决于$ \ {r_j^{（k）}的值$ \ {（k）}：然后，我们表明，提供的$ g（v_n，e_n）$在当地的弱感中收敛到明显的加尔顿 - 沃特森过程，对于任何固定的$ k $，可以耦合在$ v_n $中均匀选择的顶点的过程的动态，以一个固定的$ k $ cople $ \ \ \ \ {\ nathcal {\ nathcal {r} {r} _ \ seramets^equyq}：c}：0. 0. 0. 0.（r） K \} $在有限标记的Galton-Watson树上构建。此外，我们得出了足够的条件，在该条件下，$ \ Mathcal {r}^{（k）} _ \ emptySet $收敛为$ k \ to \ infty $，以随机变量$ \ nathcal {r}^*$可以通过吸引内源性解决方案来吸引内源性的分支机构分布固定分布的固定分配等式来表征，这些{r}^*$可以表征。我们的框架也可以应用于过程$ \ {{\ bf r}^{（k）}：k \ geq 0 \} $，其唯一的随机性源来自图形$ g（v_n，e_n）$的实现。
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 $G(V_n, E_n)$.
