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：暂无翻译