Robustness of interdependent networks under targeted attack

When an initial failure of nodes occurs in interdependent networks, a cascade of failure between the networks occurs. Earlier studies focused on random initial failures. Here we study the robustness of interdependent networks under targeted attack on high or low degree nodes. We introduce a general technique and show that the {\it targeted-attack} problem in interdependent networks can be mapped to the {\it random-attack} problem in a transformed pair of interdependent networks. We find that when the highly connected nodes are protected and have lower probability to fail, in contrast to single scale free (SF) networks where the percolation threshold $p_c=0$, coupled SF networks are significantly more vulnerable with $p_c$ significantly larger than zero. The result implies that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes that have been found useful to significantly improve robustness of single networks.

💡 Research Summary

The paper investigates how interdependent networks—systems in which two or more networks rely on each other for functionality—respond to targeted attacks that preferentially remove nodes based on their degree. While earlier work on cascading failures in such systems largely assumed random node removal, real‑world threats often focus on high‑degree “hub” nodes or, conversely, on low‑degree nodes that may be less protected. The authors develop a unified analytical framework that maps any degree‑biased attack onto an equivalent random‑failure problem on a transformed pair of networks.

The key technical step is to combine the original degree distribution (P_i(k)) of each network (i) with an attack probability function (w_i(k)) that encodes the likelihood of removing a node of degree (k). By redefining the effective degree distribution as (\tilde{P}i(k)=\frac{w_i(k)P_i(k)}{\sum{k’} w_i(k’)P_i(k’)}), the targeted‑attack scenario becomes mathematically identical to a random removal process on a network whose topology is described by (\tilde{P}_i(k)). This transformation allows the authors to reuse the well‑established generating‑function formalism for percolation on random graphs, now applied to the coupled system.

In the interdependent setting, the removal of a node in one layer instantly disables its counterpart in the other layer, creating a cascade. The authors write coupled self‑consistency equations for the size of the giant mutually connected component (GMCC) in each layer: \