Simultaneous global and local clustering in multiplex networks with covariate information

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.

💡 Research Summary

The paper introduces the Hierarchical Multiplex Stochastic Blockmodel (HMPSBM), a Bayesian framework designed to uncover both global (across‑layer) and local (within‑layer) community structures in multiplex networks while incorporating node covariate information. A multiplex network is represented as a set of L directed layers sharing the same N nodes. Each layer ℓ is modeled by an independent stochastic block model (SBM): given layer‑specific community assignments z_{ℓi} and a symmetric connection probability matrix ρ, the adjacency entries A^{ℓ}{ij} follow Bernoulli(ρ{z_{ℓi}z_{ℓj}}). The entries of ρ receive independent Beta(α₀,β₀) priors, and the same ρ is used across all layers, allowing an unbounded number of latent groups.

Global clustering is introduced through a node‑specific infinite‑dimensional probability vector τ_i, which determines the categorical distribution of a global label w_i ∈ ℕ. τ_i is constructed via a probit stick‑breaking process: τ_{ik}=Φ(x_iᵀφ_k)∏_{ℓ<k}