This note corrects an error in two related proofs of consistency of community detection: under stochastic block models by Bickel and Chen [Proc. Natl. Acad. Sci. USA 106 (2009) 21068-21073] and under degree-corrected stochastic block model by Zhao, Levina and Zhu [Ann. Statist. 40 (2012) 2266-2292].