  In recent years there has been an increased interest in statistical analysis
of data with multiple types of relations among a set of entities. Such
multi-relational data can be represented as multi-layer graphs where the set of
vertices represents the entities and multiple types of edges represent the
different relations among them. For community detection in multi-layer graphs,
we consider two random graph models, the multi-layer stochastic blockmodel
(MLSBM) and a model with a restricted parameter space, the restricted
multi-layer stochastic blockmodel (RMLSBM). We derive consistency results for
community assignments of the maximum likelihood estimators (MLEs) in both
models where MLSBM is assumed to be the true model, and either the number of
nodes or the number of types of edges or both grow. We compare MLEs in the two
models with other baseline approaches, such as separate modeling of layers,
aggregating the layers and majority voting. RMLSBM is shown to have advantage
over MLSBM when either the growth rate of the number of communities is high or
the growth rate of the average degree of the component graphs in the
multi-graph is low. We also derive minimax rates of error and sharp thresholds
for achieving consistency of community detection in both models, which are then
used to compare the multi-layer models with a baseline model, the aggregate
stochastic block model. The simulation studies and real data applications
confirm the superior performance of the multi-layer approaches in comparison to
the baseline procedures.
