Contagion dynamics on growing simplicial complex networks via generalized degree vectors

To reveal the influence of group interactions on contagion, and to accurately prevent and regulate contagion, this paper investigates the SIS contagion dynamics in complex networks with high-order interactions. First, this paper considers the synergistic effects of the first-order and second-order generalized degree of each node and proposes a novel algorithm to construct growing second-order simplicial complex networks based on generalized degree vectors, named GDV-GSC algorithm. The first-order and second-order generalized degree distributions of the networks newly achieved show heterogeneity. Then, using the microscopic Markov chain approach (MMCA), some SIS MMCA dynamics equations in the GDV-GSC network have been obtained. Monte Carlo (MC) results and MMCA numerical simulations confirm that the MMCA theoretical results are feasible and effective, and also indicate that the parameters for generating GDV-GSC network have limited effect on the contagion spreading when the effective contagion rate is either very low or very high. Further, a new method that is heterogeneous mean field approximation based on simplicial complex (called SC-HMF) to obtain epidemic threshold expressions is put forward. The MC simulation results verify the accuracy of the epidemic threshold expression and the occurrence of a bistability phenomenon, where the absorbing state and the bursting state coexist within a certain range of effective infectivity rates. Moreover, the second-order simplicial complex structures in the GDV-GSC network play a key role in determining the epidemic threshold.