The robustness of a network of networks (NON) under random attack has been studied recently [J. Gao et al., Phys. Rev. Lett. 107, 195701 (2011)]. Understanding how robust a NON is to targeted attacks is a major challenge when designing resilient infrastructures. We address here the question how the robustness of a NON under targeted attack is affected by targeted attack on high or low degree nodes. We introduce a targeted attack probability function that is dependent upon node degree, and study the robustness of two types of NON under targeted attack: (i) a tree of n fully interdependent Erdős-Rényi or scale-free networks, and (ii) a starlike network of n partially interdependent Erdős-Rényi networks. For any tree of n fully interdependent Erdős-Rényi networks and scale-free networks under targeted attack, we find that the network becomes significantly more vulnerable when nodes of higher degree have higher probability to fail. When the probability that a node will fail is proportional to its degree, for a NON composed of Erdős-Rényi networks we find analytical solutions for the mutual giant component P∞ as a function of p, where 1 − p is the initial fraction of failed nodes in each network. We also find analytical solutions for the critical fraction pc, which causes the fragmentation of the n interdependent networks, and for the minimum average degreekmin below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdős-Rényi networks under targeted attack, we find the critical coupling strength qc for different n. When q > qc, the attacked system undergoes an abrupt first order type transition. When q ≤ qc, the system displays a smooth second order percolation transition. We also evaluate how the central network becomes more vulnerable as the number of networks with the same coupling strength q increases. The limit of q = 0 represents no dependency, and the results are consistent with the classical percolation theory of a single network under targeted attack.
A case of random failure is applied to the above work, whereas in REF , the authors focused on targeted attack on a network of networks (NON).
Robustness of network of networks under targeted attack
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Mathematics Medicine