Latent space models for grouped multiplex networks
Alexander Kagan, Peter W. MacDonald, Elizaveta Levina, Ji Zhu
TL;DR
GroupMultiNeSS addresses the challenge of uncovering group-level latent structure in clustered multiplex networks by decomposing node latent positions into a shared basis $V$, group-specific $W_k$, and layer-specific $U_{k\ell}$ components, with $X_{k\ell} = [V \; W_k \; U_{k\ell}]$ and $\\Theta_{k\ell}$ admitting a decomposed form via $I_{p,q}$ blocks. The authors develop a convex two-stage estimation procedure with nuclear-norm penalties, provide identifiability guarantees up to indefinite orthogonal rotations, and prove Gaussian-edge-consistency results with explicit error bounds, supported by synthetic experiments. They demonstrate improved modeling accuracy when accounting for group structure and apply the method to Parkinson's disease brain networks, revealing biologically interpretable group differences in latent space. The approach offers a scalable, flexible framework for analyzing complex hierarchical multiplex networks and can be extended to broader hierarchical, sub-Gaussian, and directed settings.
Abstract
Complex multilayer network datasets have become ubiquitous in various applications, including neuroscience, social sciences, economics, and genetics. Notable examples include brain connectivity networks collected across multiple patients or trade networks between countries collected across multiple goods. Existing statistical approaches to such data typically focus on modeling the structure shared by all networks; some go further by accounting for individual, layer-specific variation. However, real-world multilayer networks often exhibit additional patterns shared only within certain subsets of layers, which can represent treatment and control groups, or patients grouped by a specific trait. Identifying these group-level structures can uncover systematic differences between groups of networks and influence many downstream tasks, such as testing and low-dimensional visualization. To address this gap, we introduce the GroupMultiNeSS model, which enables the simultaneous extraction of shared, group-specific, and individual latent structures from a sample of networks on a shared node set. For this model, we establish identifiability, develop a fitting procedure using convex optimization in combination with a nuclear norm penalty, and prove a guarantee of recovery for the latent positions as long as there is sufficient separation between the shared, group-specific, and individual latent subspaces. We compare the model with MultiNeSS and other models for multiplex networks in various synthetic scenarios and observe an apparent improvement in the modeling accuracy when the group component is accounted for. Experiment with the Parkinson's disease brain connectivity dataset demonstrates the superiority of GroupMultiNeSS in highlighting node-level insights on biological differences between the treatment and control patient groups.