Targeted attack on networks coupled by connectivity and dependency links

Coupled systems used to increase capacity were shown beneficial as long as it does not open pathways to cascades. Previous studies on the robustness of coupled networks except for interdependent networks are almost the cases of random attack. Many challenges remain exist in targeted-attack problem of coupled networks. Since nodes within coupled networks show different functions for each network, this paper both analytically and numerically analyzed the robustness of coupled networks under three types of targeted attacking strategies, including attacking on nodes by considering internal and external degree, internal degree only, and external degree only. For coupled network with both interdependent and interconnected links, all degree distributions of intra- and inter-connectivity links are Poissonian, we find the system undergoes from second to first order phase transition as coupling strength q increases. The fraction of nodes in the giant component P∞ at stable state, the critical phase transition threshold pc (first order threshold pcI and second order threshold pcII), and the critical point (pc,qc) separating the first and second order phase transitions are analytically obtained for three types of attacking strategies with attacking probability parameter α=0,1. We also find the system becomes more vulnerable as the average degree of intra-links k̄ or inter-links K̄ decreases. Therefore, the minimum average degrees k̄min and K̄min to maintain the system stable are obtained for the case of α=0,1. Moreover, we discussed three typical cases of coupled networks, interdependent networks (K̄=0), interacting networks (q=0) and bipartite network (k=0, q=0), the analytical expressions of P∞, pc and (pc,qc) (only for interdependent) are given respectively. Besides, we study the equivalence relations between interdependent networks and coupled networks with connectivity and dependency links for the same pc. The results imply that we can adjust the parameters q, k̄ and K̄ to make the two systems have the same robustness. Moreover, the results imply that one should not only protect nodes with high degree of intra-links or inter-links, but also defend nodes, the sum of whose degree of intra-links and inter-links is big, can avoid the system becoming vulnerable.