Robustness of multiple interdependent networks under shell attack

Shell structure can help us understand the transport properties of the network clearly, and also identify the influential spreaders in complex networks. The robustness of networks suffering shell attacks is a topic of great interest since it is an important issue affecting many real-world networks. We study the robustness of two coupled network systems under shell attack, where multiple support-dependence relations are randomly built between nodes of networks. By using the general function of the degree distribution and taking Erdos-Renyi (ER) networks as an example, we numerically and analytically study the system robustness under shell attack to find the percolation threshold R-c(A). AS we remain the mean degree of networks k (k = k(A) = k(B)) unchanged, let the ((k) over tilde a supplort degree of a node in one network, (k) over tilde = (k) over tilde (A) = (k) over tilde (B)) decrease, R-c(A) decreases too and the robustness of networks is reinforced. By exchanging the positions of k and (k) over tilde, we can get the similar conclusion. The results suggest that the robustness of the networks with multiple support-dependence links under shell attack relies not only on connectivity links of networks but also on support links.