HyperGen: Learning Structure-Aware Spectral Flows for Hypergraph Generation

Hypergraphs provide a natural representation for high-order relations, but real-world hypergraph data is often difficult to share or collect due to privacy, access, and annotation constraints. Hypergraph generation is therefore useful for constructing surrogate high-order relational data. Existing generators, however, typically build hyperedges through structural heuristics, propagation rules, or projected graph spaces, and thus do not directly model the incidence relation between nodes and hyperedges. Spectral representations offer a principled way to encode global and local relational patterns, but hypergraph spectra are intrinsically non-invertible: a spectral representation does not determine a unique discrete incidence structure. We propose HyperGen, a structure-aware generative framework that models hypergraph generation as continuous dynamics in a phase-enhanced spectral latent space. HyperGen constructs a Hermitian node--hyperedge spectral operator with learnable incidence phases, learns a conditional vector field that transports noise toward target spectral embeddings, and decodes the generated latents into node--hyperedge incidence matrices through a learned spectrum-to-structure mapping with denoising. Experiments on benchmark hypergraphs show that HyperGen more accurately recovers node--hyperedge incidences and better preserves spectral and geometric characteristics than existing baselines.