用结构化扩散模型直接生成高阶关系图，更真实还原复杂连接模式。
Hypergraph Generation via Structured Stochastic Diffusion
- 基于松弛关联矩阵设计结构化随机扩散过程，保留高阶交互特征。
- 在理想条件下实现生成精确性，且反向过程有闭式解可直接求解。
- 相比强基线模型生成质量显著提升，适合需要真实高阶关系的场景。
超图能刻画高阶交互关系，但真实超图生成困难，因关联性、超边大小异质性及重叠结构难以通过成对简化准确捕捉。我们提出 \\(\text{HEDGE}\\)，一种直接定义在松弛关联矩阵上的生成模型，采用结构化随机扩散机制。前向过程结合超图特有双向热算子与奥恩斯坦-乌伦贝克成分，在保持数据附近结构感知去噪的同时，获得明确的高斯终态分布。在观测超图条件下，该前向过程为线性高斯型，因此条件均值、协方差、得分函数及反向漂移目标均可显式表达。我们通过回归精确条件目标，学习一个置换等变的状态仅反向漂移场，并从高斯基底分布出发模拟学习到的反向时间随机微分方程生成样本。我们在理想状态仅设置下证明了精确性，并提供有限时间稳定保证；实证表明其生成质量优于多个强基线模型。
原文摘要 · Abstract (English)
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.
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