# Research on the Robustness of Interdependent Networks under Localized Attack
## 3 Results and Analysis
### Infrastructure Network  
To analyze the robustness of interdependent infrastructure networks when facing disasters, such as earthquakes, the robustness of an interdependent cyber-physical network, consisting of an infrastructure network, the western states power grid network of the United States, as shown in Figure 5a, and its control network, are analyzed under PDLA in this section. The node number of the power grid network \(N_{\text{power}}\) = 4941, the edge number is 6594, the average degree \(<\)\(K_{\text{power}}\)\(>\)\(\approx\) 2.67, the clustering coefficient \(C\) = 0.107, and the average shortest path length \(APL\) = 18.989. Figure 4 shows the illustration and degree of distribution of the power grid network. Due to the lack of computer control network data, we use a BA scale-free network instead [15,48], with \(N_{\text{BA}}\) = 4941 and \(<\)\(K_{\text{BA}}\)\(>\)\(\approx\) 4, to construct power-BA interdependent networks.  
Figure 6 shows the robustness of the power grid and power-BA under four attack strategies, and critical thresholds are listed in Table 3. Additionally, we can observe the following situations:  
Figure 5: (**a**) Illustration of power grid, the node size represents degree of node, and the layout is not equivalent to the actual location of power grid nodes. (**b**) Degree of distribution of the power grid, the inset is the degree of distribution with the \(y\)-axis in the logarithmic scale.  
Figure 6: Cascading failures of an isolated power grid and an interdependent power-BA network under different attack strategies. (**a**) Cascading failures of the power grid averaged over 100 realizations; (**b**) cascading failures of power-BA averaged over 100 realizations; (**c**) cascading failures of the power grid under one realization; and (**d**) cascading failures of power-BA under one realization.  
Firstly, because the degree of distribution and clustering coefficient of the power grid show small world characteristics, the order of attack strategies according to the damage of the power grid is the same as that of the WS network, TA > RA > TLA > PDLA, as shown in Figure 6a. The robustness of the power grid under TA is poor, very similar to the BA network, because of the existence of several hub nodes. Moreover, Figure 5a shows that the power grid has several clusters connected by many nodes arranged in lines. With the expansion of the attack range, once the connecting nodes are attacked, the power grid will break into several independent clusters and many nodes will fail, leading to a poor robustness of the power grid under RA.  
Secondly, compared with the results of WS-WS and BA-BA in Figure 4c,f, the robustness of the interdependent power-BA networks in Figure 6b show the same, but more severe, decreasing trend with the increase of the attack area. In the scenario of a targeted attack, the entire interdependent networks will completely collapse with a critical threshold \(p_{\rm c}\) around 0.04. Even in the PDLA scenario, the critical threshold \(p_{\rm c}\) is 0.42, much less than that in WS-WS and BA-BA.  
Lastly, from Figure 6c,d we can see that the percolation of the isolated power grid shows a second-order; however, the percolation of power-BA shows a first-order transition under all four attack strategies, and the critical threshold \(p_{\rm c}\) becomes very small, as listed in Table 3. This shows that, like the artificial network in Section 3.1, the correlation between the control network and infrastructure network also increases the risk of interdependent systems under attack.