Phase Transitions in Community Detection: A Solvable Toy Model

Abstract

Recently, it was shown that there is a phase transition in the community detection problem. This transition was first computed using the cavity method, and has been proved rigorously in the case of $q=2$ groups. However, analytic calculations using the cavity method are challenging since they require us to understand probability distributions of messages. We study analogous transitions in so-called "zero-temperature inference" model, where this distribution is supported only on the most-likely messages. Furthermore, whenever several messages are equally likely, we break the tie by choosing among them with equal probability. While the resulting analysis does not give the correct values of the thresholds, it does reproduce some of the qualitative features of the system. It predicts a first-order detectability transition whenever $q>2$, while the finite-temperature cavity method shows that this is the case only when $q>4$. It also has a regime analogous to the "hard but detectable" phase, where the community structure can be partially recovered, but only when the initial messages are sufficiently accurate. Finally, we study a semisupervised setting where we are given the correct labels for a fraction $\rho$ of the nodes. For $q>2$, we find a regime where the accuracy jumps discontinuously at a critical value of $\rho$.

6 pages, 6 figures

References in corpus (5)

• Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications
• Phase transition in the detection of modules in sparse networks
• Graph spectra and the detectability of community structure in networks
• (Un)detectable cluster structure in sparse networks
• Global disorder transition in the community structure of large-q Potts systems

Cited by in corpus (10)

• Phase transitions in semisupervised clustering of sparse networks
• Community detection in networks with unequal groups
• Universality of the stochastic block model
• Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors
• Finite size analysis of the detectability limit of the stochastic block model
• Detectability thresholds of general modular graphs
• Inference of hidden structures in complex physical systems by multi-scale clustering
• Large Deviations of Semi-supervised Learning in the Stochastic Block Model
• Local Algorithms for Block Models with Side Information
• Mutual Information in Community Detection with Covariate Information and Correlated Networks