Clique entropy and dynamics of clique-driven weighted hypernetworks

A hypernetwork describes an interaction involving more than two nodes. It has been applied in the fields of society, technology, biology, transportation, and port terminal management systems. In this work, first, we give the definitions of a clique, higher-order degree of cliques, entropy of cliques, and entropy of clique strength degrees. We unify the concepts of a clique in complex networks, a simplex in higher-order networks, and a tightly related set in hypernetworks. Second, in order to study the clique-driven weighted hypernetwork, an equilibrium factor evolution model of hypernetworks is proposed. The difference equation analysis method is introduced into network research, which makes the analysis of the network more rigorous. It avoids the assumption of node degree continuity in the traditional analysis network. By using Poisson process theory, the analytical expression of the cumulative distribution of higher-order degrees is obtained. Third, by considering the weight changes of the hyperedge including the selected clique, a new weighted evolving model in hypernetworks is proposed. Based on the conclusion of the equilibrium factor evolution model, the weighted hypernetwork evolution model is analyzed, and the analytic expressions of the higher-order degree distribution and higher-order strength density function of the weighted hypernetwork are obtained. The results show that the higher-order degree distribution and the density functions of higher-order strengths of the model are generalized power-law functions. The higher-order degree of clique-driven hypernetworks shows a scale-free phenomenon, and the less clique elements, the more obvious the scale-free phenomenon emerges. Theoretical findings are validated through computer simulations.