Detectability thresholds and optimal algorithms for community structure in dynamic networks

TL;DR

This paper establishes the fundamental detectability threshold for community detection in dynamic networks and introduces two algorithms—belief propagation and spectral clustering—that achieve optimal performance down to this limit.

Contribution

It derives the exact detectability threshold in dynamic stochastic block models and proposes two algorithms that are proven to be optimal at this threshold.

Findings

Exact detectability threshold derived as a function of change rate and community strength

Belief propagation achieves asymptotically optimal accuracy

Spectral clustering based on linearized BP equations performs optimally

Abstract

We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several existing models), we are able to derive the detectability threshold exactly, as a function of the rate of change and the strength of the communities. Below this threshold, we claim that no algorithm can identify the communities better than chance. We then give two algorithms that are optimal in the sense that they succeed all the way down to this limit. The first uses belief propagation (BP), which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the BP equations. We verify our analytic and algorithmic results via numerical simulation, and close with a brief discussion of extensions and open questions.