# Research on the Robustness of Interdependent Networks under Localized Attack
## 3 Results and Analysis
### Effect of Coupling Preference and Coupling Strength  
As mentioned in the introduction, the research on the robustness of interdependent networks usually takes into account different subnetworks, attack strategies, and coupling modes. Sections 3.1 and 3.2 have analyzed the robustness of several interdependent networks from the point of view of attack strategy. To ensure the integrity of the study, this section is carried out from the perspective of coupling, including the effect of coupling preference and coupling strength on interdependent networks under PDLA. Moreover, since many real networks show small-world or scale-free features [49], WS and BA network models are chosen to construct interdependent networks in this section, and analyze the impact of coupling preference and coupling strength on the robustness of interdependent networks under PDLA. Specifically, we use the same procedures as in Section 3.1 to analyze the robustness of fully-coupled WS-WS and BA-BA interdependent networks with assortative coupling (AC), disassortative coupling (DC), and random coupling (RC). Partially-interdependent system with random coupling, shown as Figure 2a, is also analyzed. Figure 7 shows the cascading failures of interdependent networks with different coupling preferences. Table 4 lists the order of coupling preferences based on the robustness of the interdependent networks under four attack strategies. The results indicate that, in three kinds of attack strategies, LA, RA, and TLA, coupling preferences do have an impact on the robustness of interdependent networks. Such as in the case of targeted attack, both interdependent WS-WS and BA-BA networks show the best robustness under disassortative coupling; when facing a random attack, the robustness under assortative coupling is the best, while the robustness under disassortative coupling is the worst. Especially in BA-BA networks, the difference among different coupling preference is much more obvious, as shown in Figure 7b. However, in the PDLA scenario, the robustness of interdependent networks with different coupling preferences is very similar. This suggests coupling preference is not the main factor that affects the robustness of interdependent networks under PDLA, which is different from the other three attack strategies. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Attack Strategy** & **TA** & **RA** & **TLA** & **PDLA** \\ \hline power grid & 0.2 & \(\approx\)0.6 & \(\approx\)1 & 1 \\ power-BA & 0.04 & 0.18 & 0.3 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Critical threshold \(p_{\rm c}\) of power-BA under different attack strategies. Moreover, Figure 8 shows the cascading failures of interdependent networks under different coupling strengths under PDLA. In each subgraph, we plot the failure curves with specific node sizes and average degree. Through the comparison of Figure 8a,b and Figure 8d,e, we see that node size of the subnetwork has little effect on the robustness, and there is a negative correlation between coupling strength and robustness: the stronger the coupling strength is, the worse the robustness is. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Attack Strategy** & **TA** & **RA** & **TLA** & **PDLA** \\ \hline WS-WS & RC \(<\) AC \(<\) DC & DC \(\approx\) RC \(<\) AC & RC \(<\) DC \(\approx\) AC & RC \(\approx\) DC \(\approx\) AC \\ BA-BA & AC \(<\) RC \(<\) DC & DC \(<\) RC \(<\) AC & AC \(<\) RC \(<\) DC & RC \(\approx\) DC \(\approx\) AC \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Order of coupling preference based on the robustness of interdependent networks. Figure 8: Cascading failure of interdependent networks with different coupling strengths under PDLA. Each point is averaged over 100 independent realizations. The relative size of giant component is defined here as \(S=G_{A}/N_{A}\).