Cascading load model in interdependent networks with coupled strength

Considering the coupled strength between interdependent networks, we introduce a new method to define the initial load on an edge and propose a cascading load model in interdependent networks. We explore the robustness of the interdependent networks against cascading failures by two measures, i.e., the critical threshold βc quantifying the whole robustness of the interdependent networks to avoid the emergence of cascading failure, and the new proposed smallest capacity threshold βc,s quantifying the degree of the worst damage of the interdependent networks. We numerically find that the AL (high-degree nodes in network A connect high-degree ones in network B) link between two networks can greatly enhance the robust level of the interdependent networks against cascading failures. Especially we observe that the values of βc in the interdependent networks with both the DL (high-degree nodes in network A connect low-degree ones in network B) link and the RL (nodes in network A randomly connect ones in network B) link increase monotonically with the coupled strength, while the values of βc,s in the interdependent networks with three types of link patterns almost monotonically decreases with the coupled strength. In the interdependent networks with the AL, the value of βc first decreases and then increases with the coupled strength. We further explain this interesting phenomenon by a simple graph. In addition, we study the influence of the coupled strength on the efficiency of two attacks to destroy the interdependent networks. We find that, when the coupled strength between two networks is weaker, attacking the edges with the lower load is more easier to trigger the cascading propagation than attacking the nodes with the higher load, however, when the coupled strength in two networks is stronger, the case is on the contrary. Finally, we give reasonable explanations from the local perspective of the total capacity of all neighboring edges of a failed edge.