Tightly related sets and collective degree distribution on hypernetworks

A hypernetwork is the most general and unconstrained mathematical model used to describe the increasingly complex relationships among things in the real world. The scale-free property of a hypernetwork is an important research topic. The research on the scale-free characteristics of hypernetworks is based on the distribution characteristics of the hyperdegree. However, the definition of hyperdegree focuses on a single node. To explore the characteristics of the groups widely existing in hypernetworks and the structural advantages of hypernetworks in representing complex systems, this study extends the concepts of hyperdegree and hyperdegree distribution in hypernetworks. Based on the concept of tightly related sets, the definition of collective degree is given. Then, the definition of collective degree distribution is proposed. The theoretical analysis results of the random uniform hypernetworks show that its collective degree distribution follows a Poisson distribution. The hyperdegree of the existing hypernetwork model generalized by the BA model (usually called a scale-free hypernetwork) follows a power law distribution. The simulation experiments show that its collective degree also follows a power law distribution. Research on real hypernetworks shows that the distribution characteristics of the hyperdegree cannot completely determine the distribution characteristics of the collective degree. The results of this study can enrich the content of hypergraph theory, provide valuable references for analyzing group characteristics in hypernetworks, and help broaden the application scope of network science.