Consistent Estimation of Dynamic and Multi-layer Networks

Dynamic networks where edges appear and disappear over time and multi-layer networks that deal with multiple types of connections arise in many applications. In this paper, we consider the multi-graph stochastic block model proposed by Holland et al. (1983), which serves as a foundation for both dynamic and multi-layer networks. We extend inference techniques in the analysis of single networks, namely maximum-likelihood estimation, spectral clustering, and variational approximation, to the multi-graph stochastic block model. Moreover we provide sufficient conditions for consistency of the spectral clustering and maximum-likelihood estimates. We verify the conditions for our results via simulation and demonstrate that the conditions are practical. In addition, we apply the model to two real data sets: a dynamic social network and a multi-layer social network, resulting in block estimates that reveal network structure in both cases.