Hypergraph Generation via Structured Stochastic Diffusion

TL;DR

This paper introduces \\HEDGE, a novel hypergraph generative model using structured stochastic diffusion directly on incidence matrices, capturing complex hypergraph structures more faithfully.

Contribution

The paper presents a new hypergraph generation method based on a structured stochastic diffusion process, improving over existing pairwise reduction approaches.

Findings

mpirically outperforms strong baselines in hypergraph generation quality.

Provides a closed-form solution for conditional means, covariances, scores, and reverse-drift targets.

Establishes theoretical guarantees for exactness and stability in the proposed diffusion process.

Abstract

Hypergraphs model higher-order interactions, but realistic hypergraph generation remains difficult because incidence, hyperedge-size heterogeneity, and overlap structure are not faithfully captured by pairwise reductions. We propose \HEDGE, a generative model defined directly on relaxed incidence matrices via a structured stochastic diffusion. The forward process combines a hypergraph-specific two-sided heat operator with an Ornstein--Uhlenbeck component, preserving structure-aware noising near the data while yielding an explicit Gaussian terminal law. Conditional on an observed hypergraph, this forward process is linear-Gaussian, so conditional means, covariances, scores, and reverse-drift targets are available in closed form. We therefore learn a permutation-equivariant state-only reverse-drift field in incidence space by regressing onto exact conditional targets, and generate samples by simulating a learned reverse-time SDE from the Gaussian base law. We establish exactness in the ideal state-only setting together with finite-horizon stability guarantees, and empirically show improved hypergraph generation quality relative to strong baselines.