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We consider the problem of recovering hidden communities in the LabeledStochastic Block Model (LSBM) with a finite number of clusters, where clustersizes grow linearly with the total number $n$ of items. In the LSBM, a label is(independently) observed for each pair of items. Our objective is to devise anefficient algorithm that recovers clusters using the observed labels. To thisend, we revisit instance-specific lower bounds on the expected number ofmisclassified items satisfied by any clustering algorithm. We presentInstance-Adaptive Clustering (IAC), the first algorithm whose performancematches these lower bounds both in expectation and with high probability. IACconsists of a one-time spectral clustering algorithm followed by an iterativelikelihood-based cluster assignment improvement. This approach is based on theinstance-specific lower bound and does not require any model parameters,including the number of clusters. By performing the spectral clustering onlyonce, IAC maintains an overall computational complexity of $\mathcal{O}(n\text{polylog}(n))$. We illustrate the effectiveness of our approach throughnumerical experiments.

Instance-Optimal Cluster Recovery in the Labeled Stochastic Block Model