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We study the hierarchy of communities in real-world networks under a genericstochastic block model, in which the connection probabilities are structured ina binary tree. Under such model, a standard recursive bi-partitioning algorithmis dividing the network into two communities based on the Fiedler vector of theunnormalized graph Laplacian and repeating the split until a stopping ruleindicates no further community structures. We prove the strong consistency ofthis method under a wide range of model parameters, which include sparsenetworks with node degrees as small as $O(\log n)$. In addition, unlike most ofexisting work, our theory covers multiscale networks where the connectionprobabilities may differ by orders of magnitude, which comprise an importantclass of models that are practically relevant but technically challenging todeal with. Finally we demonstrate the performance of our algorithm on syntheticdata and real-world examples.

Consistency of Spectral Clustering on Hierarchical Stochastic Block Models