## 4 Case Study
### Simulation Analysis of Irvulnerability Measurement  
Based on the simulation algorithm design of the invulnerability measurement described in the foregoing text, MATLAB (R2022b) software programming was employed to conduct simulations of four major types of disturbance events, namely external attacks, natural disasters, technical malfunctions, and fluctuations in passenger and freight traffic, on the nodes of the railway network in China. The simulation results, which illustrate the changing trends of the largest connected subgraphs of the railway transportation network under different disturbance scenarios, are presented in Figures 9 and 10. The figure reveals that the size of the largest connected subgraph declines more rapidly under deliberate disturbances, indicating that the network is particularly vulnerable to perturbations targeting critical nodes. This is attributable to the fact that certain nodes within the railway network play disproportionately important roles in maintaining overall connectivity. Once these nodes are removed, numerous pairs of nodes that were previously  
Figure 8: Network resilience reachability trend graph. “Perturbations with randomness” represents “random disturbances like natural disasters and technical malfunctions”; “Perturbations with directionality” represents “directed disturbances such as external attacks and fluctuations in passenger and cargo flow”. Figure 9: Trend chart of the maximum connectivity subgraph under natural disaster and technical fault simulation. connected via these nodes will cease to be directly linked, which leads to a substantial reduction in the network's connectivity. Random disturbances typically result in a more gradual descent in connectivity because the disturbances occur randomly. Nevertheless, as the perturbation persists, even random disturbances might eliminate some critical nodes, which likewise causes a decrease in network connectivity. During the simulation process, the issue of cascading failures is considered. Initially, the load of each node is set as its betweenness centrality value, while the capacity of each node is its load multiplied by a coefficient (here \(1+\sigma\)), where \(\sigma\) is a small positive number representing the enhancement coefficient of capacity and is set at 0.1. For deliberate disturbances, nodes are removed in descending order of betweenness centrality, whereas for random disturbances, nodes are removed arbitrarily. To substantiate our claim regarding the efficiency of the proposed load redistribution method, we conducted a comparative analysis with other commonly employed load allocation approaches: average allocation and path weight allocation. The comparative evaluation was performed using three key metrics:  
Network Preservation Rate (NPR): The ratio of nodes that remain functional after cascading failure to the total nodes before disturbance. Load Distribution Fairness (LDF): Measured using Jain's fairness index to quantify how equitably load is distributed among neighboring nodes. Computational Efficiency (CE): The average computation time required to complete the load redistribution process after each node failure. Table 4 presents the comparative results across the three load redistribution methods under different disturbance scenarios, with each value representing the average of 50 independent simulation runs. Statistical analysis using one-way ANOVA followed by [PERSON]'s post hoc tests confirmed significant differences between the three redistribution methods across all metrics (\(p<0.001\)). The residual capacity-based method consistently outperformed the other approaches in terms of the Network Preservation Rate (F = 37.45, \(p<0.001\)) and Load Distribution Fairness (F = 52.18, \(p<0.001\)), particularly under targeted attack scenarios where the differential impact was most pronounced. The average allocation method, while computationally less intensive, demonstrated the poorest performance in preserving network integrity during cascading failures.