## 4 Case Study
### Simulation Analysis of Irvulnerability Measurement
This is primarily because it fails to account for the heterogeneous capacity characteristics of  
Figure 10: Trend chart of the maximum connectivity subgraph under external attack and passenger and cargo flow fluctuation simulation. neighboring nodes, resulting in secondary failures when nodes with already high loads receive additional burden. The path weight allocation method showed an intermediate performance in terms of network preservation but required significantly higher computational resources (F = 43.62, \(p\) < 0.001) due to the necessity of recalculating all shortest paths after each node failure. This becomes particularly problematic in large-scale networks such as China's railway system. Our residual capacity-based approach demonstrated an optimal balance between computational efficiency and network preservation capability. The method's superior performance can be attributed to its ability to distribute excess load proportionally to each node's available capacity, thereby minimizing the likelihood of triggering cascading failures. Although slightly more computationally intensive than the average allocation approach, the significant improvements in network preservation (+16.2% on average) and Load Distribution Fairness (+38.5% on average) justify its implementation. After the removal of each node, it is verified whether the remaining nodes have a load exceeding their capacity. If an overload is detected, the excess load is redistributed to adjacent nodes. This redistribution may cause neighboring nodes to exceed their own capacities, thereby initiating further rounds of cascading failures. This process persists until no node has a load exceeding its capacity. After each perturbation and cascading failure, the size of the maximum connected subgraph of the network is computed and updated, which serves as a key metric for gauging the network's resilience. It is processed as the resilience measurement result and presented in Figure 11.  
Figure 11 indicates that the network is more susceptible when exposed to highly targeted disturbances. In comparison, while disruptions caused by natural disasters, technical failures, and fluctuations in passenger and freight traffic also lead to a decline in \begin{table}
\begin{tabular}{c c c c c} \hline
**Redistribution Method** & **Disturbance Type** & \begin{tabular}{c} **Network** \\ **Preservation Rate** \\ \end{tabular} & \
\begin{tabular}{c} **Load Distribution** \\ **Fairness** \\ \end{tabular} &
\begin{tabular}{c} **Computational** \\ **Efficiency (ms)** \\ \end{tabular} \\ \hline Residual Capacity-Based (Proposed) & External Attack & 0.682 & 0.874 & 23.7 \\ Average Allocation & External Attack & 0.587 & 0.631 & 19.2 \\ Path Weight Allocation & External Attack & 0.635 & 0.792 & 31.5 \\ Residual & & & & \\ Capacity-Based (Proposed) & Natural Disaster & 0.831 & 0.868 & 22.3 \\ Average Allocation & Natural Disaster & 0.782 & 0.625 & 18.8 \\ Path Weight Allocation & Natural Disaster & 0.806 & 0.787 & 30.8 \\ Residual & & & & \\ Capacity-Based (Proposed) & Technical Failure & 0.845 & 0.863 & 22.1 \\ Average Allocation & Technical Failure & 0.798 & 0.622 & 18.9 \\ Path Weight Allocation & Technical Failure & 0.817 & 0.781 & 30.6 \\ Residual & & & & \\ Capacity-Based (Proposed) & Flow Fluctuation & 0.714 & 0.871 & 23.5 \\ Average Allocation & Flow Fluctuation & 0.621 & 0.627 & 19.1 \\ Path Weight Allocation & Flow Fluctuation & 0.673 & 0.788 & 31.2 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparative performance of different load redistribution methods. network connectivity, the rate of decline is comparatively slower. This suggests that the network possesses a certain degree of resilience against random or non-targeted disruptions. Moreover, Figure 11 visually and explicitly reflects the effect of the occurrence of cascading failures on the invulnerability of the network. As disturbances accumulate over time, even non-deliberate or naturally occurring failures may impact critical nodes.