Recovering a hidden community beyond the Kesten–Stigum threshold 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 weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime $n^{1-o(1)} \leq K \leq o(n).$ A critical parameter is the effective signal-to-noise ratio $\lambda=K^2(p-q)^2/((n-K)q)$, with $\lambda=1$ corresponding to the Kesten-Stigum threshold. We show that a belief propagation algorithm achieves weak recovery if $\lambda>1/e$, beyond the Kesten-Stigum threshold by a factor of $1/e.$ The belief propagation algorithm only needs to run for $\log^\ast n+O(1) $ iterations, with the total time complexity $O(|E| \log^*n)$, where $\log^*n$ is the iterated logarithm of $n.$ Conversely, if $\lambda \leq 1/e$, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying power iteration to the non-backtracking matrix of the graph is shown to attain weak recovery if and only if $\lambda>1$. 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 $p/q$.