The Deep Latent Position Block Model for the Clustering of Nodes in Multi-graphs

Network data capture relationships among actors across multiple contexts, often forming clusters of individuals. These relationships frequently involve multiple types of interactions, requiring multidimensional networks, or multi-graphs, to fully capture their complexity. Latent position models (LPMs) embed nodes based on connection probabilities but cannot uncover heterogeneous clustering structures such as disassortative patterns. In contrast, stochastic block models (SBMs) excel at clustering but lack interpretable latent representations. To address these limitations, the deep latent position block model (Deep-LPBM) was introduced to provide both clustering and continuous latent space representations in unidimensional networks. In this paper, we extend this approach to multidimensional networks by introducing the deep latent position block model for multidimensional networks (Deep-LPBMM). Deep-LPBMM integrates block modeling and latent embedding across multiple interaction types, allowing nodes to partially belong to several groups, thereby capturing overlapping clustering structures more effectively. Our model employs a deep variational autoencoder with graph convolutional networks (GCNs) for each layer and a multilayer perceptron to merge the latent representations into a unified embedding that represents partial cluster membership probabilities, enabling both effective clustering and enhanced visualization.