In recent years there has been an increased interest in statistical analysisof data with multiple types of relations among a set of entities. Suchmulti-relational data can be represented as multi-layer graphs where the set ofvertices represents the entities and multiple types of edges represent thedifferent 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 restrictedmulti-layer stochastic blockmodel (RMLSBM). We derive consistency results forcommunity assignments of the maximum likelihood estimators (MLEs) in bothmodels where MLSBM is assumed to be the true model, and either the number ofnodes or the number of types of edges or both grow. We compare MLEs in the twomodels with other baseline approaches, such as separate modeling of layers,aggregating the layers and majority voting. RMLSBM is shown to have advantageover MLSBM when either the growth rate of the number of communities is high orthe growth rate of the average degree of the component graphs in themulti-graph is low. We also derive minimax rates of error and sharp thresholdsfor achieving consistency of community detection in both models, which are thenused to compare the multi-layer models with a baseline model, the aggregatestochastic block model. The simulation studies and real data applicationsconfirm the superior performance of the multi-layer approaches in comparison tothe baseline procedures.
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