Robustness of higher-order interdependent networks

In recent years, the research of multilayer interdependent networks has become a hotspot in complex networks. However, most of the study is limited to describing pairwise interactions. The systems in the real world are usually networks with higher-order interactions consisting of three or more units, such as epidemic transmission and cooperative communication networks. To better reflect the complex networks in the real world, this paper introduces the higher-order structures in the network, that is, simplicial complexes. In this paper, we construct a theoretical model of a two-layer partial dependence network with simplicial complexes in which failures between nodes occur through the synergistic effects of pairwise and higher-order interactions. In this model, removing a node will cause all other nodes in the same simplex to be removed, and due to the dependency between the two networks, the failure of the node will spread through dependency links between the two networks. This process will occur recursively and eventually lead to the cascading process. We introduce percolation theory to study the robustness of the network and give the theoretical solutions of different properties of the network, such as the size of the mutually connected giant component (MCGC), percolation threshold, etc. We find that the density of the triangle and the dependent strength between the two networks affect the percolation behaviours of the network together. When the density of the triangle exceeds a certain value, the network shows a double transition.