Hypergraph Generation via Structured Stochastic Diffusion

Hypergraphs model complex systems where interactions involve more than two entities, but generating realistic hypergraphs has been challenging because existing methods fail to capture key properties like incidence patterns, hyperedge-size heterogeneity, and overlap structure. Traditional pairwise reductions lose these higher-order features, limiting their use in domains like social network analysis, biology, and recommendation systems.

Christopher Nemeth introduces HEDGE, a generative model built directly on relaxed incidence matrices via a structured stochastic diffusion. The forward process uses a novel two-sided heat operator tailored to hypergraphs, combined with an Ornstein-Uhlenbeck process, allowing structure-aware noising that converges to an explicit Gaussian base law. Because the forward process is linear-Gaussian, conditional distributions—including means, covariances, and score functions—are available in closed form. HEDGE learns a permutation-equivariant reverse-drift field by regressing onto these exact targets, then generates samples by simulating a learned reverse-time SDE. The paper provides theoretical guarantees of exactness in the ideal state-only setting and finite-horizon stability, and empirical results show improved hypergraph generation quality over strong baselines.