## 2 Analytical Framework
### Topological Transformation of Geometric Transportation Network  
This study utilized the natural street topological transformation method proposed by [PERSON] and [PERSON] [42] to convert the geometric transportation network and AADT data into a topological representation. Natural street typologies were generated by considering the continuity of street segments and the angle changes between intermediate road segments [43]. As demonstrated by [PERSON] and [PERSON] [21], this transformation method accurately represents human movement, traffic flow behavior, and the hierarchical nature of urban transportation networks. The study employed the Axwoman Extension in ArcGIS, a tool introduced by [PERSON] et al. [44], to facilitate the topological transformation. Within the Axwoman Extension, natural roads are created by tracking the angle changes of each road segment. To transform the geometric road network into a natural street topological representation, the process begins by splitting the network into individual road segments at each intersection point, enabling the measurement of angle changes for each segment. Natural streets are then generated by applying a predefined threshold angle change, ensuring an accurate representation of the network's topological structure.  
In this study, we set the threshold for angle change at 20\({}^{\circ}\), merging intermediate segments into a single segment if the angle change was below this threshold, while segments with angle changes above the threshold remained separate. Both the geometric road network and the AADT dataset of the study area initially consisted of 1210 individual road segments. After applying the 20\({}^{\circ}\) threshold, the natural street network was reduced to 323 street segments, which were then used to evaluate the vulnerability and traffic flow impact of the transportation network under critical-link disruptions. The rationale for selecting a 20\({}^{\circ}\) angle threshold for topological transformation is justified through both empirical evidence from prior studies and a sensitivity analysis of alternative thresholds. The 20\({}^{\circ}\) threshold has been widely adopted in previous studies, such as by [PERSON] et al. [29], [PERSON] et al. [45], and [PERSON] and [PERSON] [35], which have demonstrated its effectiveness in capturing the continuity of human-perceived roadways while minimizing excessive network segmentation. Similarly, [PERSON] et al. [46] employed comparable thresholds to align natural street networks with human navigation patterns and spatial cognition. To further validate this choice, a sensitivity analysis was conducted by testing thresholds ranging from 10\({}^{\circ}\) to 60\({}^{\circ}\). The results revealed that thresholds lower than 20\({}^{\circ}\) (e.g., 10\({}^{\circ}\)) overgeneralized the network, losing critical local variations, while higher thresholds (e.g., 30-60\({}^{\circ}\)) led to excessive fragmentation, complicating analysis and interpretation. The 20\({}^{\circ}\) threshold struck an optimal balance, preserving meaningful structural details and maintaining key topological properties such as connectivity, centrality, and clustering. This combination of empirical support from prior research and the findings from the sensitivity analysis robustly justifies the use of the 20\({}^{\circ}\) threshold for topological transformation in this study.