Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown

TL;DR

This paper proves that a modified adjacency-spectral partitioning method can consistently identify communities in stochastic block models even when the number of blocks and the rank are unknown, demonstrating robustness to model mis-specification.

Contribution

It introduces a spectral partitioning procedure that only requires an upper bound on the rank, extending its applicability to multi-modal and directed/undirected graphs.

Findings

The modified spectral method is consistent with only an upper bound on the rank.

The procedure remains robust under model mis-specification.

Extension to multi-modal and directed/undirected graphs is successful.

Abstract

For random graphs distributed according to a stochastic block model, we consider the inferential task of partioning vertices into blocks using spectral techniques. Spectral partioning using the normalized Laplacian and the adjacency matrix have both been shown to be consistent as the number of vertices tend to infinity. Importantly, both procedures require that the number of blocks and the rank of the communication probability matrix are known, even as the rest of the parameters may be unknown. In this article, we prove that the (suitably modified) adjacency-spectral partitioning procedure, requiring only an upper bound on the rank of the communication probability matrix, is consistent. Indeed, this result demonstrates a robustness to model mis-specification; an overestimate of the rank may impose a moderate performance penalty, but the procedure is still consistent. Furthermore, we extend this procedure to the setting where adjacencies may have multiple modalities and we allow for either directed or undirected graphs.