Kronecker Generative Models for Power-Law Patterns in Real-World Hypergraphs

Do real-world hypergraphs obey any patterns? Are power laws fundamental in hypergraphs as they are in real-world graphs? What generator can reproduce these patterns? A hypergraph is a generalization of a conventional graph, and it consists of nodes and hyperedges, with each hyperedge joining any number of nodes. Hypergraphs are adept at representing group interactions where two or more entities interact simultaneously, such as collaborative research and group discussions. In a wide range of real-world hypergraphs, we discover power-law or log-logistic distributions in eight structural properties. To simulate these observed patterns, we introduce HyRec, a tractable and realistic generative model leveraging the Kronecker product. We mathematically demonstrate that HyRec accurately reproduces both the patterns we observed and typical evolutionary trends found in real-world hypergraphs. To fit the parameters of HyRec to large-scale hypergraphs, we design SingFit, a fast and space-efficient algorithm successfully applied to eleven real-world hypergraphs with up to one million nodes and hyperedges. This paper makes the following contributions: (a) Discoveries: we identify multiple patterns that real-world hypergraphs obey, (b) Model: we propose HyRec, a tractable and realistic model capable of reproducing real-world hypergraphs efficiently (spec., with fewer than 1,000 parameters) with the support of SingFit, and (c) Proofs: we prove that HyRec adheres to these patterns.