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

The stochastic block model for one community with parameters n,K, p, and q is considered: K out of n vertices are in the community; two vertices are connected by an edge with probability p if they are both in the community and with probability q otherwise, where p > q > 0 and p/q is assumed to be bounded. An estimator based on observation of the graph G = (V,E) is said to achieve weak recovery if the mean number of misclassified vertices is o(K) as n → ∞. A critical role is played by the effective signal-to-noise ratio λ = K2(p− q)2/((n−K)q). In the regime K = Θ(n), a näıve degree-thresholding algorithm achieves weak recovery in O(|E|) time if λ → ∞, which coincides with the information theoretic possibility of weak recovery. The main focus of the paper is on weak recovery in the sublinear regime K = o(n) and np = n. It is shown that weak recovery is provided by a belief propagation algorithm running for log∗(n)+O(1) iterations, if λ > 1/e, with the total time complexityO(|E| log∗ n). Conversely, no local algorithm with radius t of interaction satisfying t = o( logn log(2+np) ) can asymptotically outperform trivial random guessing if λ ≤ 1/e. 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 provide weak recovery if λ ≤ 1.