Preferential Attachment Hypergraph Model With Randomized Hyperedge Count and Size

Hypergraphs provide a natural framework for modeling higher-order interactions among nodes, with real-world hypergraph data often exhibiting weighted and heterogeneous structures. In this article, we propose a preferential attachment hypergraph model that incorporates randomness in two key components: the number of hyperedges a node connects to upon arrival, and the size of each hyperedge. By allowing these quantities to follow arbitrary probability distributions, the model generalizes prior fixed-parameter settings and better reflects empirical observations. We analytically derive the asymptotic degree distribution under mild assumptions, showing that it follows a power-law whose exponent is governed by the distribution of hyperedge sizes. Moreover, we show that the distribution of hyperedge counts significantly influences the degree behavior in the small-degree regime, where deviations from ideal power-law patterns are often seen in real-world networks. We also demonstrate that the overall degree distribution can be expressed as a mixture of fixed-count degree distributions. Simulation results and real data analyses support the theoretical findings and highlight the practical relevance of the proposed model.