Statistical methods for hypergraphs: a parameter estimator, a model selection, and a comparative test

Graphs have long been used to model complex systems, but real-world networks often exhibit fluctuations that challenge traditional graph-based methods. Random graph models and spectral techniques have been employed for statistical analysis. However, even these approaches are limited to dyadic relationships, whereas real-world systems often involve more complex interactions. Hypergraphs, which generalize graphs by allowing edges to connect multiple nodes, offer a more accurate representation of such complexity. Therefore, we propose a framework to statistically analyze real-world systems based on the hypergraph’s adjacency matrix spectrum. First, we introduce the Kullback–Leibler divergence to compare the spectra of two hypergraphs. We then develop statistical methods for hypergraph analysis, including a parameter estimator, a model selection approach, and a method to test whether two or more hypergraphs were generated by the same process (i.e. the same model and parameter set). Simulation experiments demonstrate the efficacy of our methods. Finally, we apply our approach to real-world hypergraphs as an illustrative example.