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Community detection is a fundamental problem in network analysis with manymethods available to estimate communities. Most of these methods assume thatthe number of communities is known, which is often not the case in practice. Westudy a simple and very fast method for estimating the number of communitiesbased on the spectral properties of certain graph operators, such as thenon-backtracking matrix and the Bethe Hessian matrix. We show that the methodperforms well under several models and a wide range of parameters, and isguaranteed to be consistent under several asymptotic regimes. We compare thismethod to several existing methods for estimating the number of communities andshow that it is both more accurate and more computationally efficient.

Estimating the number of communities in networks by spectral methods