Dependency-based targeted attacks in interdependent networks

Modern large engineered network systems normally work in cooperation and incorporate dependencies between their components for purposes of efficiency and regulation. Such dependencies may become a major risk since they can cause small-scale failures to propagate throughout the system. Thus, the dependent nodes could be a natural target for malicious attacks that aim to exploit these vulnerabilities. Here we consider a type of targeted attack that is based on the dependencies between the networks. We study strategies of attacks that range from dependency-first to dependency-last, where a fraction $1-p$ of the nodes with dependency links, or nodes without dependency links, respectively, are initially attacked. We systematically analyze, both analytically and numerically, the percolation transition of partially interdependent networks, where a fraction $q$ of the nodes in each network are dependent on nodes in the other network. We find that for a broad range of dependency strength $q$, the “dependency-first” attack strategy is actually less effective, in terms of lower critical percolation threshold $p_c$, compared with random attacks of the same size. In contrast, the “dependency-last” attack strategy is more effective, i.e., higher $p_c$, compared with a random attack. This effect is explained by exploring the dynamics of the cascading failures initiated by dependency-based attacks. We show that while “dependency-first” strategy increases the short-term impact of the initial attack, in the long term the cascade slows down compared with the case of random attacks and vice versa for “dependency-last.” Our results demonstrate that the effectiveness of attack strategies over a system of interdependent networks should be evaluated not only by the immediate impact but mainly by the accumulated damage during the process of cascading failures. This highlights the importance of understanding the dynamics of avalanches that may occur due to different scenarios of failures in order to design resilient critical infrastructures.

Figure 1

A schematic plot of partially interdependent networks, under three types of attacks. (a) Random attack: Nodes are attacked at random. (b) Degree-based targeted attack: The order of attacked nodes is determined by the number of connected nodes in the individual networks. (c) Dependency-based targeted attacks: In “dependency-first” attack, nodes are attacked first if their failure will cause node failure in the other network, due to dependency relations. In “dependency-last” attack, these nodes are attacked last.

Figure 2

Percolation transition of partially interdependent ER networks ($k_A=k_B=4$) under targeted attacks. Three levels of dependency strength are shown: (a) $q=0.9$, (b) $q=0.8$, and (c) $q=0.6$. Upper panels: The size of network $A$ at steady state, ${\psi }_{\infty }$, after an initial attack of ($1-p$) of the nodes is shown versus the fraction of the remaining nodes $p$. Random attack ($\beta =0$) is shown in black (middle). “Dependency-first” ($\beta =1$) is shown in red [right in (a) and (b) and left in (c)]. “Dependency-last” ($\beta =-1$) is shown in blue [left in (a) and (b) and right in (c)]. Circles and solid lines represent the simulation and analytical results, respectively. Middle and lower panels show simulation results of the second-largest component of network $A, {\psi }_{\infty 2}$, and the number of iterations in the cascade (NOI), respectively, versus $p$. The numerical results represent single realization of interdependent networks, each with $N={10}^6$ nodes.

Figure 3

The effect of coupling strength $q$ on the stability of interdependent networks under dependency-based and degree-based attacks. (a) Numerical and analytical results of $p_c$ versus $q$, for partially interdependent ER networks ($k_A=k_B=4$), each with $N=3\times {10}^5$ nodes. Results represent an average over 10 realizations. Random attack (black plus signs) is compared with dependency-first ($\beta =1$, red circles) and dependency-last ($\beta =-1$, blue triangles) targeted attacks. Symbols and lines indicate numerical and analytical results, respectively. Solid lines and dashed lines indicate first-order and second-order transitions, respectively, for each curve. The red dashed line corresponds to the region where all dependency nodes are initially attacked. Gray symbols represent four intermediate cases $\beta =\pm 1/3, \beta =\pm 2/3$ in simulation. (b) The same as (a) for the cases of high-degree first (red, upper) and low-degree first (blue, lower) targeted attacks. The ratio between the critical threshold $p_c$ of each of the attacks and the one for same-size random attack, $p_c/p_{c,\mathrm{random}}$ versus $q$, (c) for dependency-based attacks and (d) for degree-based attacks.

Figure 4

(a) The ratio between the giant component size of $A$ under dependency-first attack ($\beta =1$) and that under random attack, ${\psi }_t/{\psi }_{t,\mathrm{random}}$ versus $t$, analytical results. Partially interdependent ER networks. Here $k_A=k_B=4, q$ changes from 0.76 to 0.82, and $p$ is taken as the critical value $p_c$ in theory. Red lines with circles or crosses show the two cases with $q<q^{*(\beta =1)}$, i.e., $q=0.76$ and $q=0.78$, respectively. Blue lines with squares or diamonds show the two cases with $q>q^{*(\beta =1)}$, i.e., $q=0.80$ and $q=0.82$, respectively. (b) The same as (a) but for the dependency-last attack ($\beta =-1$). Red lines with circles or crosses show the two cases with $q<q^{*(\beta =-1)}$, i.e., $q=0.72$ and $q=0.74$, respectively. Blue lines with squares or diamonds show the two cases with $q>q^{*(\beta =-1)}$, i.e., $q=0.76$ and $q=0.78$, respectively.

Figure 5

Percolation transition in network $B$ of partially interdependent ER networks ($k_A=k_B=4.0$) under targeted attacks. Three levels of dependency strength are shown: (a) $q=0.9$, (b) $q=0.8$, and (c) $q=0.6$. Upper panels: The size of network $B$ at steady state, ${\varphi }_{\infty }$, after an initial attack of ($1-p$) of the nodes is shown versus the fraction of the remaining nodes $p$. Random attack ($\beta =0$, black) is compared with “dependency-first” ($\beta =1$, red) and “dependency-last” ($\beta =-1$, blue). Middle and lower panels show simulation results of the second-largest component of network $B, {\varphi }_{\infty 2}$, and the NOI, respectively, versus $p$. The numerical results represent single realization of interdependent networks, each with $N={10}^6$ nodes.

Figure 6

The effect of coupling strength $q$ on the stability of interdependent ER networks with varying mean degree under dependency-based attacks. (a) Numerical results of $p_c$ versus $q$, for partially interdependent ER networks ($k_A=k_B=3.0$), each with $N=3\times {10}^5$ nodes. Results represent an average over 10 realizations. Random attack (black) is compared with dependency-first ($\beta =1$, red) and dependency-last ($\beta =-1$, blue) targeted attacks. Solid lines and dashed lines indicate first-order and second-order transitions, respectively, for each curve. (b) The ratio between the critical threshold $p_c$ of each of the attacks and the one for same-size random attack, $p_c/p_{c,\mathrm{random}}$ as a function of $q$, for partially interdependent ER networks with $k_A=k_B=3.0$. [(c) and (d)] The same as in (a) and (b) but for $k_A=k_B=5.0$.

Figure 7

The effect of coupling strength $q$ on the stability of interdependent SF networks under dependency-based attacks. (a) Numerical results of $p_c$ versus $q$, for partially interdependent SF networks ($\gamma =2.5, k_A=k_B=4.0$), each with $N=3\times {10}^5$ nodes. Results represent an average over 10 realizations. Random attack (black) is compared with dependency-first ($\beta =1$, red) and dependency-last ($\beta =-1$, blue) targeted attacks. Solid lines and dashed lines indicate first-order and second-order transitions, respectively, for each curve. The red dashed line corresponds to the region where all dependency nodes are initially attacked. (b) The ratio between the critical threshold $p_c$ of each of the attacks and the one for same-size random attack, $p_c/p_{c,\mathrm{random}}$, as a function of $q$.

Figure 8

The effect of coupling strength $q$ on the stability of interdependent networks with degree-degree correlation under dependency-based attacks. (a) Numerical results of $p_c$ versus $q$, for partially interdependent ER networks ($k_A=k_B=4$) with degree-degree correlation, each with $N=3\times {10}^5$ nodes. Results represent an average over 10 realizations. Random attack (black) is compared with dependency-first ($\beta =1$, red) and dependency-last ($\beta =-1$, blue) targeted attacks. Solid lines and dashed lines indicate first-order and second-order transitions, respectively, for each curve. (b) The ratio between the critical threshold $p_c$ of each of the attacks and the one for same-size random attack, $p_c/p_{c,\mathrm{random}}$, versus $q$.

Figure 9

The effect of coupling strength $q$ on the stability of interdependent networks with degree-dependency correlation under dependency-based attacks. (a) Numerical results of $p_c$ versus $q$, for partially interdependent ER networks ($k_A=k_B=4$) with degree-dependency correlation, each with $N=3\times {10}^5$ nodes. Results represent an average over 10 realizations. Random attack (black) is compared with dependency-first ($\beta =1$, red) and dependency-last ($\beta =-1$, blue) targeted attacks. Solid lines and dashed lines indicate first-order and second-order transitions, respectively, for each curve. (b) The ratio between the critical threshold $p_c$ of each of the attacks and the one for same-size random attack, $p_c/p_{c,\mathrm{random}}$, versus $q$.