Simultaneous global and local clustering in multiplex networks with covariate information

TL;DR

The paper tackles the challenge of uncovering both cross-layer (global) and layer-specific (local) community structure in multiplex networks while incorporating nodal covariates. It introduces the HMPSBM, a Bayesian nonparametric model that jointly infers an unbounded global clustering via probit stick-breaking with GEM priors and within-layer SBMs, sharing a common connectivity structure across layers. A scalable variational inference framework with mean-field factorization and truncation ($M_z$, $M_w$) enables efficient fitting to large networks, and extensive simulations show robust recovery of global and layer-level groups under diverse conditions; the FAO trade network application demonstrates meaningful latent structure aligned with economic interpretation and covariate-driven refinements. Overall, HMPSBM advances multilayer network analysis by coupling global covariate-informed clustering with flexible layer-level community detection, providing a practical tool for revealing complex cross-layer patterns in economic and biological networks; code is made available.

Abstract

Understanding both global and layer-specific group structures is useful for uncovering complex patterns in networks with multiple interaction types. In this work, we introduce a new model, the hierarchical multiplex stochastic blockmodel (HMPSBM), that simultaneously detects communities within individual layers of a multiplex network while inferring a global node clustering across the layers. A stochastic blockmodel is assumed in each layer, with probabilities of layer-level group memberships determined by a node's global group assignment. Our model uses a Bayesian framework, employing a probit stick-breaking process to construct node-specific mixing proportions over a set of shared Griffiths-Engen-McCloseky (GEM) distributions. These proportions determine layer-level community assignment, allowing for an unknown and varying number of groups across layers, while incorporating nodal covariate information to inform the global clustering. We propose a scalable variational inference procedure with parallelisable updates for application to large networks. Extensive simulation studies demonstrate our model's ability to accurately recover both global and layer-level clusters in complicated settings, and applications to real data showcase the model's effectiveness in uncovering interesting latent network structure.

Figures (8)

• Figure 1: Illustration of the parameters and directed acyclic graph for the HMPSBM in \ref{['eqn:model_adjacency']}-\ref{['eqn:model_gamma_k']}.
• Figure 2: Toy example of the group structure of nodes in a multiplex network with $L=3$ layers corresponding to the illustration in Figure \ref{['fig:model_schematic']}, with the grey global group as circles and the white as diamonds. There are two global groups, indicated by the shape of the nodes, and three layer groups, indicated by the colours. Probabilities of layer group assignments given the global groups are indicated in the row vectors.
• Figure 3: Boxplots of the NMI score with increasing $\alpha$ for the global and layer groups under the settings of Section \ref{['subsec:sim3']}.
• Figure 4: Boxplots of the NMI score with varying number of nodes, $N$, and varying numbe of samples, $M$, for the global and layer groups under the settings of Section \ref{['subsec:sim4']}.
• Figure 5: Distribution of layer-level groups in each global group using different features.