Traditional graph representations are insufficient for modelling real-world phenomena involving multi-entity interactions, such as collaborative projects or protein complexes, necessitating the use of hypergraphs. While hypergraphs preserve the intrinsic nature of such complex relationships, existing models often overlook temporal evolution in relational data. To address this, we introduce a first-order autoregressive (i.e. AR(1)) model for dynamic non-uniform hypergraphs. This is the first dynamic hypergraph model with provable theoretical guarantees, explicitly defining the temporal evolution of hyperedge presence through transition probabilities that govern persistence and change dynamics. This framework provides closed-form expressions for key probabilistic properties and facilitates straightforward maximum-likelihood inference with uniform error bounds and asymptotic normality, along with a permutation-based diagnostic test. We also consider an AR(1) hypergraph stochastic block model (HSBM), where a novel Laplacian enables exact and efficient latent community recovery via a spectral clustering algorithm. Furthermore, we develop a likelihood-based change-point estimator for the HSBM to detect structural breaks within the time series. The efficacy and practical value of our methods are comprehensively demonstrated through extensive simulation studies and compelling applications to a primary school interaction data set and the Enron email corpus, revealing insightful community structures and significant temporal changes.

翻译：传统图表示方法不足以建模涉及多实体交互的现实世界现象（如协作项目或蛋白质复合物），因此需要使用超图。虽然超图保留了此类复杂关系的内在特性，但现有模型往往忽略关系数据中的时间演化。为解决这一问题，我们针对动态非均匀超图提出一阶自回归（即AR(1)）模型。这是首个具备可证明理论保证的动态超图模型，通过控制持续性与变化动态的转移概率，明确定义了超边存在性的时间演化。该框架为关键概率特性提供闭式表达式，支持具有一致误差界与渐近正态性的直接最大似然推断，并包含基于置换的诊断检验。我们还提出AR(1)超图随机块模型（HSBM），其中新型拉普拉斯矩阵通过谱聚类算法实现精确高效的潜在社区恢复。此外，我们为HSBM开发了基于似然的变化点估计器，用于检测时间序列中的结构突变。通过大量模拟研究，以及对小学互动数据集和安然邮件语料库的实证应用，我们全面验证了所提方法的效能与实用价值，揭示了富有洞察力的社区结构与显著的时间变化。