Let $N$ components be partitioned into two communities, denoted ${\cal P}_+$ and ${\cal P}_-$, possibly of different sizes. Assume that they are connected via a directed and weighted Erd\"os-R\'enyi random graph (DWER) with unknown parameter $ p \in (0, 1).$ The weights assigned to the existing connections are of mean-field type, scaling as $N^{-1}$. At each time unit, we observe the state of each component: either it sends some signal to its successors (in the directed graph) or remains silent otherwise. In this paper, we show that it is possible to find the communities ${\cal P}_+$ and ${\cal P}_-$ based only on the activity of the $N$ components observed over $T$ time units. More specifically, we propose a simple algorithm for which the probability of {\it exact recovery} converges to $1$ as long as $(N/T^{1/2})\log(NT) \to 0$, as $T$ and $N$ diverge. Interestingly, this simple algorithm does not require any prior knowledge on the other model parameters (e.g. the edge probability $p$). The key step in our analysis is to derive an asymptotic approximation of the one unit time-lagged covariance matrix associated to the states of the $N$ components, as $N$ diverges. This asymptotic approximation relies on the study of the behavior of the solutions of a matrix equation of Stein type satisfied by the simultaneous (0-lagged) covariance matrix associated to the states of the components. This study is challenging, specially because the simultaneous covariance matrix is random since it depends on the underlying DWER random graph.

翻译：假设 $N$ 个组分被划分为两个社区，记为 ${\cal P}_+$ 和 ${\cal P}_-$，其规模可能不同。假定它们通过一个参数未知 $ p \in (0, 1)$ 的有向加权 Erd\"os-R\'enyi 随机图（DWER）连接。现有连接上分配的权重为平均场类型，按 $N^{-1}$ 缩放。在每个时间单位，我们观测每个组分的状态：它要么向其（在有向图中的）后继者发送信号，要么保持静默。本文证明，仅基于 $N$ 个组分在 $T$ 个时间单位内观测到的活动，即可找到社区 ${\cal P}_+$ 和 ${\cal P}_-$。具体而言，我们提出一种简单算法，当 $T$ 和 $N$ 趋于无穷且 $(N/T^{1/2})\log(NT) \to 0$ 时，其实现{\it 精确恢复}的概率收敛于 $1$。值得注意的是，该简单算法无需任何关于其他模型参数（如边概率 $p$）的先验知识。我们分析的关键步骤是推导与 $N$ 个组分状态相关联的单位时间滞后协方差矩阵在 $N$ 趋于无穷时的渐近近似。该渐近近似依赖于对 Stein 型矩阵方程解行为的研究，该方程由与组分状态相关联的同步（0-滞后）协方差矩阵所满足。此项研究具有挑战性，特别是因为同步协方差矩阵是随机的，因其依赖于底层的 DWER 随机图。