Consistency of Spectral Clustering in Sparse Stochastic Block Models

We analyze the performance of spectral clustering for community extraction in sparse stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of magnitude of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. The key components of our analysis are a careful perturbation analysis of the principal subspaces of the adjacency matrix and a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.