Optimal Cluster Recovery in the Labeled Stochastic Block Model

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (labeled SBM) 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 address the finer and more challenging question of determining, under the general LSBM and for any $s=o(n)$, the set of parameters such that there exists a polynomial-time clustering algorithm with at most $s$ misclassified items in average. We prove that a necessary and sufficient condition to get $s$ misclassified items in average is $\frac{n D(\alpha,p)}{ \log (n/s)} \ge 1$, where $D(\alpha,p)$, referred to as the {\it divergence}, is an appropriately defined function of the parameters $p=(p(i,j,\ell), i,j, \ell)$, and $\alpha$ defining the sizes of the clusters. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit. The algorithm runs in $O(n^2\log(n))$ time. The analysis presented in this paper allows us to recover existing results for asymptotically accurate and exact cluster recovery in the SBM, but has much broader applications. For example, it implies that the minimal number of misclassified items under the LSBM considered scales as $n\exp(-nD(\alpha,p)(1+o(1)))$.