Universal Phase Transition in Community Detectability under a Stochastic Block Model

TL;DR

This paper establishes a universal phase transition threshold for community detectability in stochastic block models using spectral modularity, showing that detection success depends on the inter-community connection probability relative to within-community probabilities.

Contribution

The paper proves the existence of a universal asymptotic phase transition threshold for community detection under a stochastic block model, independent of community size ratios.

Findings

The phase transition threshold is at p* = sqrt(p1 * p2).

Successful detection occurs when inter-community probability p < p*.

The threshold is validated through simulations for moderate community sizes.

Abstract

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 $\sqrt{p_1\cdot p_2}$, where $p_i~(i=1,2)$ 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. Using the derived expression for the phase transition threshold we propose an empirical method for estimating this threshold from real-world data.