Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel

Abstract

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 multilayer 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. 2(1− )N R m I (m)

The community structure of functional brain networks exhibits scale-specific patterns of variability across individuals and time

“…Other methods, such as stochastic blockmodels [63], can detect more general classes of communities. Future work could build on recent applications of blockmodeling to brain network data [64,65] while taking advantage of multi-layer formulations to study multi-subject cohorts [66][67][68]. In addition, the modularity maximization framework is subject to socalled resolution limits [69,70] that, for a given set of parameters, {γ, ω}, render it incapable of resolving communities below some characteristic size.…”

Clustering Network Layers with the Strata Multilayer Stochastic Block Model

“…Given the usefulness of SBMs for the understanding of node organization in single-layer networks, it is important to extend SBMs to the multilayer framework, and indeed this direction of research is receiving growing attention [7], [31], [32], [33], [34]. In this context, the general assumption is that there are shared patterns in community structure across the layers of a multilayer network, and the goal is to define and identify a stochastic block model that captures this structure.…”

“…[7], [31], [32], the authors studied situations in which many layers follow from a single SBM. In these instances, it is possible to obtain improved inference of the SBM parameters by incorporating multiple samples from a single model.…”

“…[7] was followed up by Ref. [31], wherein the authors addressed issues that can arise for the model when K and/or L is large, or if the network is sparse. They proposed a modified model called the restricted multilayer stochastic block model (rMLSBM).…”

“…[32] is similar to Refs. [7] and [31] in that the authors seek to leverage information from all layers by considering the joint distribution of layers. Using this, they estimated quantities such as the marginal probabilities of node assignments to communities and the edge probabilities within and between groups.…”