Robustness of circularly interdependent networks

Circularly interdependent multilayer networks widely exist in nature, such as the food webs and world trade networks, where each layer depends on one another and the dependencies among all layers form a directed loop. In this paper we study the robustness of the circularly interdependent multilayer networks with three or more layers under random node attacks by using the percolation method. We propose an analytical framework to predict the critical threshold and size of giant component in the steady state, where the analytical results agree well with the simulation results. We focus on two types of dependencies, the one-to-one and one-to-many dependency. In the one-to-one interdependent networks, there exists a tricritical point of the node dependence strength at which the phase transition switches between first and second order, which is independent of the average degree and the number of layers. When the dependence strength between node pairs is large, increasing the number of layers leads to the increase of percolation threshold. In the three-layer one-to-many interdependent networks with strong coupling strength, under different inter-layer average degrees, the system undergoes the first-order transition and the robustness of the system increases with the inter-layer average degrees. Our work provides a quantitative understanding of the robustness of circularly interdependent networks.