Autoregressive hypergraph

Traditional graph representations are insufficient for modelling real-world phenom ena 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 tem poral evolution in relational data. To address this, we introduce a first-order autore gressive (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 ex pressions 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 effi cient latent community recovery via a spectral clustering algorithm. Furthermore, we develop a likelihood-based change-point estimator for the HSBM to detect struc tural breaks. 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.