Community detection for binary graphical models in high dimension

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 (DWER) random graph 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 \modif{step}, 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 \modif{ two simple methods, an aggregated method and a spectral method, whose {\it misclassification rates} vanish as long as $T \gg N$ (up to log terms). This condition is proved to be near-optimal in the minimax sense. Moreover, under the stronger condition $T \gg N^2$ (up to log terms), the aggregated method is shown to achieve {\it exact recovery} with probability tending to $1$. } Interestingly, these simple \modif{methods} do not require any prior knowledge of the other model parameters (e.g. the edge probability $p$). The key step in our analysis is to derive an asymptotic approximation of the 1-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 \modif{Stein-type} matrix equation satisfied by the simultaneous (0-lagged) covariance matrix associated to the states of the components. This study is challenging, especially because the simultaneous covariance matrix is random since it depends on the underlying DWER random graph.