Fiona Skerman: Community detection and the overlap gap property

Abstract:

We study the theoretical limits of local algorithms based on the modularity score to recover planted communities in random graphs.

The modularity score \(q_{\mathcal{A}}(G)\) gives a measure of how well the vertex partition \(\mathcal{A}\) divides the graph \(G\) into `communities'; and is central to the most popular algorithms used to cluster real network data. We consider recovery in the Stochastic Block Model (SBM) where the graph \(G\) has a planted \(k\)-block partition where vertices within the same block connect with probability \(p\) while vertices in different blocks connect with probability \(q\), independently across pairs of vertices.

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. We establish that modularity exhibits OGP on the SBM. This rules out recovery in the SBM by a class of local algorithms based on the modularity score and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established on a `planted' rather than `null' model.

As part of our proof we extend a result by Bickel and Chen [PNAS 2009]; who established that with high probability, a modularity optimal partition of SBM is few local moves away from the planted partition; we extend this to non-optimal partitions whose score is sufficiently close to the optimum score.

Joint work with Shankar Bhamidi, David Gamarnik, Remco van der Hofstad, Nelly Litvak, Pawel Pralat and Yasmin Tousinejad.