Optimal Cluster Recovery in the Labeled Stochastic Block Model

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number $K$ of clusters of sizes linearly growing with the global population of items $n$. Every pair of items is labeled independently at random, and label $\ell$ appears with probability $p(i,j,\ell)$ between two items in clusters indexed by $i$ and $j$, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most $s$ misclassified items in average under the general LSBM and for any $s=o(n)$, which solves one open problem raised in \cite{abbe2015community}. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within $O(n \mbox{polylog}(n))$ computations and without the a-priori knowledge of the model parameters.