Consistent estimation of dynamic and multi-layer block models

TL;DR

This paper analyzes the asymptotic properties of spectral clustering and MLE estimators for multi-graph stochastic block models, providing conditions for consistency and practical algorithms, with applications to real dynamic and multi-layer networks.

Contribution

It extends theoretical analysis of SBM estimators to multi-graph models, deriving consistency conditions and proposing a scalable variational MLE approximation.

Findings

Both estimators are consistent under derived conditions.

The variational MLE is computationally feasible for large networks.

Applications demonstrate practical relevance to real-world networks.

Abstract

Significant progress has been made recently on theoretical analysis of estimators for the stochastic block model (SBM). In this paper, we consider the multi-graph SBM, which serves as a foundation for many application settings including dynamic and multi-layer networks. We explore the asymptotic properties of two estimators for the multi-graph SBM, namely spectral clustering and the maximum-likelihood estimate (MLE), as the number of layers of the multi-graph increases. We derive sufficient conditions for consistency of both estimators and propose a variational approximation to the MLE that is computationally feasible for large networks. We verify the sufficient conditions via simulation and demonstrate that they are practical. In addition, we apply the model to two real data sets: a dynamic social network and a multi-layer social network with several types of relations.