## 2 Analytical Framework
### Critical-Link Attack Method  
This study introduces a method for identifying network vulnerabilities under worst-case scenarios by targeting critical links derived from topological parameters, focusing on their significance within specific topological aspects rather than overall network performance. By focusing on disruptions to each topological parameter in its specific context, this approach provides a nuanced assessment of network vulnerability. Such a perspective is crucial for evaluating the network's vulnerability from different topological viewpoints, enabling a detailed understanding of how specific parameters influence network performance. Consequently, while some segments may significantly impact network functionality due to their topological significance, others might not, even if deemed critical from a specific topological perspective.  
The method evaluates network vulnerability by simulating sequential disruptions to critical road segments identified through topological measures, proceeding iteratively in descending order of their significance. During this process, once a critical road segment is attacked (or removed from the network), it is no longer part of the network for subsequent iterations. The topological parameters are recalculated for the remaining network after each attack, and the criticality of the remaining segments is reassessed based on the updated network structure. This ensures that the most critical components identified in each iteration may differ from those in previous iterations, as the removal of a segment can alter the network's topology and redistribute the significance of the remaining segments. This dynamic reassessment captures the evolving nature of network vulnerability and ensures that the worst-case scenario is accurately represented.  
This approach captures worst-case vulnerabilities efficiently, distinguishing itself from earlier studies by [PERSON] and [PERSON] [14] and [PERSON] and [PERSON] [15], which rely on more computationally intensive and probabilistic methods. Its simplicity and reliance on minimal data enhance its feasibility for large-scale networks, particularly in underdeveloped and developing regions with limited data availability. Figure 1 illustrates the flowchart of the proposed methodology, highlighting its practicality and applicability for assessing network vulnerabilities from multiple topological perspectives.  
As shown in Figure 1, the process begins by calculating each topological parameter under undisrupted conditions. The received topological values for each road segment are then sorted in descending order to identify the maximum value (critical link segment) for each parameter. The identified segments are subjected to attacks (or dropped from the network). Subsequently, the same topological parameters are recalculated to assess the effects of the attacked segment on network performance in terms of the respective topological parameter aspect. The process continues, with the values sorted again for each segment, and the maximum topology value was captured to evaluate the network vulnerability after each attack iteration. This iterative process persists until all road segments (TSs) have been  
Figure 1: Flowchart of the critical-link failure method.  
attacked (i.e., attacked segments--ASs), such that TSs equal ASs. This method captures the worst-case transportation network vulnerability scenario and holistically assesses the network vulnerability by considering all potential individual disruptions.