Recovering a Hidden Community Beyond the Spectral Limit in $O(|E| \log^*|V|)$ Time

Community detection is considered for a stochastic block model graph of $n$ vertices, with $K$ vertices in the planted community, edge probability $p$ for pairs of vertices both in the community, and edge probability $q$ for other pairs of vertices. The main focus of the paper is on recovery of the community based on the graph $G$, with $o(K)$ misclassified vertices on average, in the sublinear regime $n^{1-o(1)} \leq K \leq o(n).$ It is shown that such recovery is attainable by a belief propagation algorithm running for $\log^\ast n+O(1)$ iterations, if $\lambda =K^2(p-q)^2/((n-K)q)$, the signal-to-noise ratio, exceeds $1/e,$ with the total time complexity $O(|E| \log^*n)$. Conversely, if $\lambda \leq 1/e$, no local algorithm can asymptotically outperform trivial random guessing. By analyzing a linear message-passing algorithm that corresponds to applying power iteration to the non-backtracking matrix of the graph, we provide evidence to suggest that spectral methods fail to recovery the community if $\lambda \leq 1.$ In addition, the belief propagation algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all $K \ge \frac{n}{\log n} \left( \rho_{\rm BP} +o(1) \right),$ where $\rho_{\rm BP}$ is a function of $p/q$.