Generative Modeling of Hypergraphs using Latent Hyperbolic Geometry

Hypergraphs encode multi-way interactions among entities arising in a broad range of real-world relational systems. Generative modeling of hypergraphs aims to capture patterns in these interactions for understanding and simulating complex relational structures. Existing approaches often face a difficult trade-off: parsimonious probabilistic models may oversimplify rich multi-way interactions, whereas expressive deep neural architectures can sacrifice interpretability and scalability. To strike a better balance between parsimony and expressivity, we introduce a novel generative model, the *Hyperbolic Hypergraph Generator*, which harnesses hyperbolic geometry as a source of expressive power while retaining a parsimonious and interpretable probabilistic latent space formulation. Theoretically, we establish identifiability of the latent geometry under a geometric compatibility condition, and characterize the gain in expressivity afforded by the hyperbolic latent space. For training and sampling, we develop *H2G*, a scalable procedure that combines likelihood-based embedding with latent space generator construction, and establish theoretical guarantees for the consistency of the generated hyperlink distribution. Extensive empirical comparisons with baseline methods show that H2G captures meaningful latent geometry and delivers strong generative performance, achieving leading results on fidelity metrics that measure the reproduction of key structural characteristics of real-world hypergraphs. Together, these theoretical and empirical results demonstrate that our model achieves an effective balance between parsimony and expressivity, while highlighting the broader potential of hyperbolic geometry for modeling complex multi-way relational systems.