Hypergraph Generation via Structured Stochastic Diffusion

Hypergraphs provide a principled framework for modeling polyadic interactions, with applications in recommendation systems, social networks, and molecular modeling. Hypergraph generation remains challenging because incidence structures are discrete, sparse, and governed by heterogeneous higher-order interactions. Existing generators often rely on implicit latent spaces or continuous incidence decoders, which provide limited mechanistic interpretation of how node-hyperedge incidences arise. To address these limitations, we propose HYVINT, an intensity-driven hypergraph generative framework. Our key innovations are twofold: (i) we develop an intensity-driven incidence formation mechanism for hypergraphs that links latent interaction strength to binary incidence, and (ii) we derive a tractable lower-bound variational estimator for learning latent representations. We provide generation error bounds with asymptotic convergence rates and empirically show that HYVINT achieves strong fidelity while maintaining substantial novelty and diversity on synthetic and real-world hypergraphs.

Hypergraph neural networks have shown powerful capability in modeling higher-order relations, yet their predictive uncertainty remains underexplored. Unlike pairwise graphs, uncertainty in hypergraphs arises not only from noisy attributes and ambiguous labels, but also from variations in node-hyperedge incidence structures and complex higher-order dependencies. Existing approaches mainly estimate uncertainty from final predictions or rely on computationally expensive ensembles and Bayesian inference, limiting their ability to capture uncertainty evolution during representation learning. In this paper, we propose Hypergraph Neural Stochastic Diffusion(HyperNSD), a stochastic differential equation framework for uncertainty estimation on hypergraphs. HyperNSD models hypergraph representations as stochastic processes evolving over node-hyperedge incidence structures. A learnable drift function captures deterministic higher-order diffusion dynamics, while a learnable stochastic forcing function characterizes structural ambiguity and representation noise. Predictive uncertainty is directly quantified through the variability of stochastic representation trajectories, providing an intrinsic uncertainty measure beyond post-hoc confidence scores. We formulate HyperNSD with neural drift and diffusion networks, enabling joint learning of prediction and uncertainty propagation. Theoretical analyses establish well posedness, perturbation stability,permutation equivariance, and numerical convergence of the proposed stochastic dynamics. Experiments on multiple hypergraph benchmarks demonstrate that HyperNSD achieves reliable uncertainty estimation for out-of-distribution and misclassification detection while preserving competitive prediction accuracy. These results provide a principled stochastic-dynamical framework for trustworthy higher-order representation learning.

Hyper-relational knowledge graphs (HKGs) effectively represent complex facts. While inferring new knowledge in HKGs is a critical problem, current methods cast it as a simple link prediction, assuming that nearly all entities and relations within a fact are known, leaving only a single blank to be filled. However, this restricted assumption may not hold in real-world scenarios in which multiple, or even all, constituent components of a fact may be missing simultaneously. To bridge this gap, we introduce a task called fact generation: generating a valid hyper-relational fact from an arbitrarily masked query, i.e., completing a partially observed fact or generating a fact from scratch. We propose KREPE, the first generative representation learning method for HKGs that learns to model the probability distributions of missing components conditioned on the local fact components and global structure of HKGs via a masked discrete diffusion. KREPE models both the intra-fact dependencies by contextual message passing and inter-fact correlations by aggregating stochastically sampled contexts. KREPE seamlessly unifies link prediction and fact generation within a single training framework, achieving state-of-the-art performance on standard HKG link prediction benchmarks and outperforming LLM-based baselines in generating novel and correct facts.