## 4 Analysis and Discussion
### Identifying the Network Performance Changes Under Critical-Link Attacks  
In this study, we used the power law package in Python to plot the distribution functions for each topological parameter, as described by [PERSON] et al. [58]. The Y-axis represents the probability of the distribution, and the X-axis represents the value of each topological parameter. As the value of X increases, the probability of X being greater than or equal to that value decreases. Figure 3 shows the CCDF of each topological parameter for undisruptive conditions and for the first 10%, 20%, and 30% of the critical-link attacks. Note that we omitted the left side of the CCDF because the main focus was on the tail of the distribution function in the context of risk management studies. As depicted in Figure 3, under undisrupted conditions (dark red curve), the network maintains high values across all measures, reflecting robust connectivity, accessibility, and flow efficiency. Thus, the tail of each distribution function becomes heavier with critical links, characterized by a greater probability and very high criticality of each respective topological parameters. As a result, compared to the tail, the head of the undisturbed conditions exhibits a lower probability and criticality of each distribution function. However, as disruptions increase to 10%, 20%, and 30%, the CCDF curves exhibit steeper declines, indicating significant losses in network performance. This demonstrates that either the significance of the critical links needed to maintain greater network performance decreases when network critical segments are disrupted, or the failure of certain roads has a lesser impact on the network when the critical segments are attacked. To validate these distributional changes at each disruptive stage, this study conducted Kolmogorov-Smirnov tests, confirming significant distribution shifts under disruptive conditions, as depicted in Figure 4.  
For instance, for SC, at 10% disruption, the curve drops more sharply, showing reduced probabilities of retaining high connectivity. For example, at a connectivity value of 0.6, the CCDF falls to approximately 0.2. With 20% and 30% disruptions, the declines become even steeper, and the CCDF truncates earlier, reflecting severe fragmentation. At 30% disruption, connectivity values above 0.4 are nearly absent, highlighting the network's fragility under escalating disruptions. For CC, at 10% disruption, the CCDF drops notably, indicating increased travel distances and reduced accessibility, with values below 0.5 having probabilities under 0.2. At 20% disruption, the decline steepens further, as centrality values fall below 0.4 for most links. By 30% disruption, the CCDF approaches zero around 0.3, reflecting severe fragmentation where accessibility across the network collapses. For BC, under 10% disruption, the curve drops earlier, with the CCDF falling below 0.2 at a betweenness value near 0.4, showing the loss of key flow-concentrated links. At 20% disruption, the drop is more pronounced, with the probability of high-centrality links declining sharply. By 30% disruption, betweenness centrality values approach zero around 0.3, highlighting the loss of critical pass-by segments that concentrate significant traffic flow within the OD trip distribution. This results in reduced flow efficiency as traffic reroutes through less central paths. Figure 3: CCDF of each topological parameter for undisruptive conditions and the first 10%, 20%, and 30% of critical-link attacks. Finally, for EC, at 10% disruption, the curve begins to deviate at higher centrality values, showing a gradual loss of influential connections. At 20% disruption, this drop becomes sharper around centrality values of 0.4, reflecting compromised connectivity among influential nodes. By 30% disruption, the CCDF exhibits a steep decline, approaching zero at values around 0.3, signifying severe fragmentation where very few links retain influence.