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 which maps the targeted-attack problem in interdependent networks to the 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.

Figure 1

(a) Dependence of $p_c$ on $\alpha$ for single and interdependent SF networks with a lower cutoff of degree $m=2$. The horizontal lines represent the upper and lower limits of $p_c$ as $\alpha =\pm \infty$. The curved dashed line represents $p_c$ for single SF networks. (b) Threshold $p_c$ vs $\lambda$ for interdependent SF networks with different $m$ and $\alpha$.

Figure 2

Values of $p_{\infty }$ vs $p$ for theory and simulation. All results are for $\alpha =1$. The symbols represent simulation data ($N={10}^6$ nodes). The solid lines are theoretical predictions based on Eq. (21) for ER networks and numerical solution of Eqs. (13) and (17) for SF networks. For the interdependent ER network case, both networks are under an initial targeted attack. For the interdependent SF network case, only one network is under an initial targeted attack. Inset: Values of $p_c$ vs average degree of ER networks with $\alpha =1$. The symbols represent simulation data, while the solid line is the theory, Eq. (22). The dashed line is $p_c$ under random attack with $\alpha =0$.