## 4 Case Study
### Statistical Analysis and Robustness Tests  
To enhance the rigor of our findings and validate the conclusions drawn from the simulation results, we conducted comprehensive statistical analyses and robustness tests. Figure 14: Network efficiency recovery diagram under natural disaster and technical fault simulation. Table 5 summarizes the key resilience metrics across different disturbance scenarios, including means, standard deviations, and confidence intervals based on 100 independent simulation runs. The statistical analysis confirms several key findings:  
1. Significant differences between disturbance types: One-way ANOVA tests revealed significant differences in accessibility (F = 147.82, \(p\) < 0.001) and invulnerability (F = 168.23, \(p\) < 0.001) across the four disturbance types. Post hoc Tukey tests confirmed that targeted disturbances (external attacks and flow fluctuations) produced significantly lower resilience values than random disturbances (natural disasters and technical failures), with \(p\) < 0.001 for all pairwise comparisons between these groups. 2. Recovery strategy effectiveness: Paired t-tests comparing recovery strategies showed that the betweenness-based strategy significantly outperformed both degree-based (t = 11.37, \(p\) < 0.001) and random strategies (t = 24.19, \(p\) < 0.001) across all disturbance types. The degree-based strategy also significantly outperformed the random strategy (t = 16.84, \(p\) < 0.001). To verify the robustness of our findings, we conducted sensitivity analyses by varying key parameters:  
1. Cascade failure threshold: We tested different capacity enhancement coefficients (\(\sigma\)) ranging from 0.05 to 0.20, finding that while the absolute values of invulnerability measures changed, the relative performance of different disturbance types remained consistent ([PERSON]'s W = 0.89, \(p\) < 0.001). 2. Network size effects: We performed simulations on random subsets of the network (50%, 70%, and 90% of nodes) to test scaling effects. The relative patterns of resilience metrics remained stable across network sizes (correlation coefficients between 0.83 and 0.91, all \(p\) < 0.001). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Disturbance Type** & **Resilience Measure** & **Mean** & **Std. Dev.** & **95\% Confidence Interval** \\ \hline \multirow{5}{*}{External Attack} & Accessibility & 0.412 & 0.038 & [0.405, 0.419] \\  & Invulnerability & 0.317 & 0.042 & [0.309, 0.325] \\  & Restorability (Betweenness) & 0.783 & 0.056 & [0.772, 0.794] \\  & Restorability (Degree) & 0.651 & 0.061 & [0.639, 0.663] \\  & Restorability (Random) & 0.425 & 0.089 & [0.407, 0.443] \\ \hline \multirow{5}{*}{Natural Disaster} & Accessibility & 0.637 & 0.044 & [0.628, 0.646] \\  & Invulnerability & 0.584 & 0.051 & [0.574, 0.594] \\  & Restorability (Betweenness) & 0.846 & 0.042 & [0.838, 0.854] \\  & Restorability (Degree) & 0.819 & 0.039 & [0.811, 0.827] \\  & Restorability (Random) & 0.726 & 0.071 & [0.712, 0.740] \\ \hline \multirow{5}{*}{Technical Failure} & Accessibility & 0.658 & 0.041 & [0.650, 0.666] \\  & Invulnerability & 0.603 & 0.047 & [0.594, 0.612] \\ \cline{1-1}  & Restorability (Betweenness) & 0.862 & 0.037 & [0.855, 0.869] \\ \cline{1-1}  & Restorability (Degree) & 0.832 & 0.041 & [0.824, 0.840] \\ \cline{1-1}  & Restorability (Random) & 0.734 & 0.068 & [0.721, 0.747] \\ \hline \multirow{5}{*}{Flow Fluctuation} & Accessibility & 0.478 & 0.046 & [0.469, 0.487] \\ \cline{1-1}  & Invulnerability & 0.349 & 0.053 & [0.339, 0.359] \\ \cline{1-1}  & Restorability (Betweenness) & 0.811 & 0.047 & [0.802, 0.820] \\ \cline{1-1}  & Restorability (Degree) & 0.702 & 0.058 & [0.691, 0.713] \\ \cline{1-1}  & Restorability (Random) & 0.463 & 0.082 & [0.447, 0.479] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical summary of resilience metrics across disturbance types. 3.