Tree-Enhanced Latent Space Models for Two-Mode Networks

Latent space models have garnered significant attention in the analysis of two-mode networks. In numerous applications, auxiliary information in the form of a hierarchical tree structure, which elucidates the interrelationships between nodes and provides extensive insights into connectivity patterns, can be easily obtained. To harness the potential of such tree-structured information, we introduce an innovative tree-enhanced latent space model (TLSM) for two-mode networks. In this framework, each node is characterized by a latent embedding vector, reparameterized as the aggregate of intermediate vectors corresponding to nodes within the tree structure. By optimizing the log-likelihood function augmented with a tree-based regularization term, the proposed model facilitates the simultaneous estimation of embedding vectors and the derivation of interpretable community structures. We have developed an efficient Alternating Direction Method of Multipliers (ADMM) algorithm to solve the resulting optimization problem. Theoretical analysis establishes the consistency of the proposed estimator under some mild conditions. Furthermore, comprehensive simulation studies and empirical applications on the Amazon review dataset substantiate the efficacy and practical relevance of the proposed model. Supplementary materials for this article are available online.