We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability p grows. This phase transition separates a sub-critical regime of small p, where modularity-based community detection successfully identifies the communities, from a super-critical regime of large p where successful community detection is impossible. We show that, as the community sizes become large, the asymptotic phase transition threshold p * is equal to √ p1p2, where pi (i = 1, 2) is the within-community edge connection probability. Thus the phase transition threshold is universal in the sense that it does not depend on the ratio of community sizes. The universal phase transition phenomenon is validated by simulations for moderately sized communities.
A universal phase transition threshold on community detectability for the modularity method under the stochastic block model is established in REF , where the asymptotic critical value depends only on the parameters of the stochastic block model and does not depend on the ratio of community sizes in the large network limit.
Universal Phase Transition in Community Detectability under a Stochastic Block Model
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