  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.
