## 3 Methodology
### Network Topology Analysis Methodology  
The network efficiency metric is a methodology for analyzing the link structure between nodes in a system, and it is widely utilized to quantitatively assess the characteristics of various complex networks. Originally introduced by [PERSON] and [PERSON] (2001), this metric has been extensively validated and applied in a wide range of disciplines [62]. [PERSON] et al. (2005) applied the efficiency metric to a real-world transportation network, the subway system, and empirically demonstrated its effectiveness in explaining both the structural properties and functional performance of the network [63]. The efficiency metric used in this study is defined as follows:  
\[Efficiency=\frac{1}{N(N-1)}{\sum_{i\
eq j}\frac{1}{d_{ij}}} \tag{1}\]  
\(N\): Number of nodes;  
\(d_{ij}\): Shortest path length between node \(i\) and \(j\). The connectivity metric, which measures how effectively all nodes within the network remain connected, is based on Menger's Theorem [64]. This metric evaluates the extent to which a network remains intact when specific nodes or edges are removed. Connectivity is expressed as follows:  
(1) Connectivity of Nodes  
\[min(|S|\ :S\subset V,\ G-S\ is\ separated) \tag{2}\]  
S: Minimum set of nodes that must be removed to disconnect the network;  
Figure 1: Overall research landscape. \(V\): Set of all nodes in graph \(G\). (2) Connectivity of Edges  
\[min(|F|\ :F\subset E,\ G-F\ is\ separated) \tag{3}\]  
\(F\): Minimum set of edges that must be removed to disconnect the network;  
\(E\): Set of all edges in graph \(G\). In addition, it defined the average shortest path length as the mean of the shortest distances between all pairs of nodes in a network. A lower average shortest path length indicates a network structure that facilitates efficient information transmission and overall higher connectivity. The shortest path length has been widely used to quantitatively evaluate the performance of transportation networks. [PERSON] and [PERSON] (2010) employed the average shortest path length to analyze subway networks in major global cities, using it to assess both network efficiency and resilience to disruptions [65]. Their study empirically demonstrated that shorter path lengths indicate better accessibility and a more robust network structure [65]. \[Average\ Shortest\ Path\ Length\ =\ \frac{1}{N(N-1)}\sum_{i\
eq j}d_{ij} \tag{4}\]  
\(N\): Number of nodes;  
\(d_{ij}\): Shortest path length between node \(i\) and \(j\). [PERSON] and [PERSON] (1998) introduced the concept of the clustering coefficient, which quantifies the proportion of triangles formed within a network [66]. The clustering coefficient measures the extent to which a given node's neighbors are interconnected, with higher values indicating a greater degree of local connectivity. [PERSON] et al. (2017) analyzed the spatial structure and transportation efficiency of road networks using average shortest path length, network centrality, and the clustering coefficient [67]. This metric is particularly useful for assessing the local structural properties of a network. \[Clustering\ Coefficient=\frac{2E_{i}}{k_{i}(k_{i}-1)} \tag{5}\]  
\(E_{i}\): Number of triangles formed by the neighbors of node \(I\);  
\(k_{i}\): Degree of node \(i\). [PERSON] (1973) introduced the concept of algebraic connectivity to assess the robustness of a network [68]. [PERSON] et al. (2014) analyzed reinforcement strategies for real-world air transportation networks using algebraic connectivity and demonstrated that this metric serves as a useful quantitative criterion for assessing network robustness [69]. Algebraic connectivity is defined as the second-smallest eigenvalue of the network's Laplacian matrix. A lower algebraic connectivity value suggests a higher likelihood of network fragmentation, indicating a greater susceptibility to disconnection:  
\[Algebric\ Connectivity\ =\ \lambda_{2} \tag{6}\]  
\(\lambda_{2}\): Second smallest eigenvalue of the network's Laplacian matrix.