Cascade phenomenon in multilayer networks with dependence groups and hierarchical structure

Many multiplex systems have hierarchical structures, such as communication networks. In this paper, we construct a cascading failure model on multilayer networks with dependence groups and hierarchical structure. We propose a new connection method between layers after generalizing the previous papers. Theoretical solutions of cascading model could be obtained by the approaches of mean-field approximation and generating function techniques. Through realizing simulations on ER and BA networks, we get the simulation results which are nearly consistent with the theoretical values. Besides, some factors affecting the network robustness are also analysed. When the initial removal probability p is small, as the group size g and the probability q1 increase, the network robustness exhibits better performance. In contrast, the network robustness decreases as g and q1 increase. The larger the failure threshold b and the recovery threshold a, the stronger the network robustness. With the decrease of the probability q2 that dependence groups connect to other nodes, the network performs better. Furthermore, our work may provide particular reference significance for enhancing the robustness of complex systems and maintaining the security of communication networks.