The stochastic block model has the overlap graph property for modularity

TL;DR

This paper demonstrates that the overlap gap property (OGP) exists for modularity-based clustering in the stochastic block model, implying limitations for local algorithms and slow mixing times.

Contribution

It establishes the presence of OGP for modularity in SBM, connecting geometric properties to algorithmic limitations and extending prior results on modularity optimality.

Findings

OGP holds for modularity in SBM, indicating local algorithm failure.

Modularity optimal partitions are close to the planted partition with high probability.

The analysis shows slow mixing times for related Markov Chain algorithms.

Abstract

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity--based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.