🤖 AI Summary
Traditional network theory models only pairwise interactions, leading to systematic representational bias when characterizing higher-order, multi-body relational structures.
Method: We propose the Exponential Random Hypergraph Model (ERHM), the first entropy-maximization framework systematically extended to higher-order interaction structures, grounded in the principle of maximum entropy. Using incidence matrices as the foundational representation, ERHM generalizes higher-order network metrics—such as hyperdegree and hyperclustering—to construct an analytically tractable, structurally interpretable null model. It generalizes both the Erdős–Rényi and configuration models to hypergraphs.
Contribution/Results: The model provides theoretical guarantees on asymptotic properties and scalable algorithms for large-scale hypergraph analysis. Empirical evaluation on real-world datasets reveals statistically significant higher-order structural patterns deviating from random baselines, demonstrating strong theoretical rigor, computational scalability, and empirical validity.
📝 Abstract
Network theory has primarily focused on pairwise relationships, disregarding many-body interactions: neglecting them, however, can lead to misleading representations of complex systems. Hypergraphs represent an increasingly popular alternative for describing polyadic interactions: our innovation lies in leveraging the representation of hypergraphs based on the incidence matrix for extending the entropy-based framework to higher-order structures. In analogy with the Exponential Random Graphs, we name the members of this novel class of models Exponential Random Hypergraphs. Here, we focus on two explicit examples, i.e. the generalisations of the Erd""os-R'enyi Model and of the Configuration Model. After discussing their asymptotic properties, we employ them to analyse real-world configurations: more specifically, i) we extend the definition of several network quantities to hypergraphs, ii) compute their expected value under each null model and iii) compare it with the empirical one, in order to detect deviations from random behaviours. Differently from currently available techniques, ours is analytically tractable, scalable and effective in singling out the structural patterns of real-world hypergraphs differing significantly from those emerging as a consequence of simpler, structural constraints.
