Cascading failures in interdependent directed networks under localized attacks

Many real-world systems interact with one another through dependency links, which reduces the system robustness. Most previous studies on the robustness of interdependent networks focus on undirected networks, and the related studies on directed networks are limited to random or targeted attacks. However, some failure scenarios cannot be described by these two kinds of attacks, such as earthquakes, floods, and epidemics, where systems are attacked in a local range. In this work, we develop a theoretical framework for analyzing the robustness of interdependent directed networks under localized attacks. We find that for degree homogeneous networks, network robustness under localized attacks is similar to that under random attacks. There are four phase transitions in the phase diagram of the network, and a four-phase transition point and a two-phase transition point are found. For degree heterogeneous networks, localized attacks are more likely to lead to collapse than random attacks. As the coupling strength between networks increases, the interdependent networks first show a second order phase transition, and then a hybrid phase transition, and a first order phase transition at last. Furthermore, as the degree heterogeneity increases, the robustness of networks first increases and then decreases, showing a local robustness maximum. The findings could help understand network robustness and enable better design of robust interdependent systems.