Competitive and Weighted Evolving Simplicial Complexes

Abstract: A simplex-based network is referred to as a higher-order network, in which describe that the interactions can include more than two nodes. Many multicomponent interactions can be grasped through simplicial complexes, which have recently found applications in social, technological, and biological contexts. The paper first proposes a competitive evolving model of higher-order networks. We introduce the difference equation analysis approach in the high-order network to make the analysis network more rigorous. It avoids the assumption that the degrees of nodes are continuous in the traditional analysis network. We obtain an analytical expression for the distribution of higher-order degrees by employing the theory of Poisson processes. The established results indicate that in a d-order network the scale-free behavior for the (d-1)-dim simplex with respect to the d-order degree is controlled by the competitiveness factor. As the competitiveness increases, the d-order degree of the (d-1)-dim simplex is bent under the logarithmic coordinates. While the e(<d-1)-dim simplex with respect to the d-order degree exhibits scale-free behavior. Second, by considering the weight changes of the neighboring simplices, as triggered by the selected simplex, a new weighted evolving model in higher-order networks is proposed. The results of the competitive evolving model of higher-order networks are used to analyze the weighted evolving model so that obtained are the analytical expressions of the higher-order degree distribution and higher-order strength density function of weighted higher-order networks. The outcomes of the simulation experiments are consistent with the theoretical analysis.