## 3 Methods
### Network Topological Metrics  
#### 3.2.1 Service Efficiency Index  
Existing studies [17; 18] commonly use complex network theory, employing topological efficiency as a performance indicator for road networks. The topological efficiency is the average efficiency of all node pairs within a network, where the efficiency of a node pair reflects their connectivity, which usually uses the inverse of their topological distance. Let \(R(G)\) denote the topological efficiency of network \(G\)[36]:  
\[R(G)=\frac{1}{N(N-1)}\sum\limits_{i\
eq j,i,j\in V}\frac{1}{d_{ij}}, \tag{1}\]  
where \(d_{ij}\) represents the shortest topological distance between node \(i\) and node \(j\), i.e., the minimum edges required to travel from node \(i\) to node \(j\). Equation (1) reflects the physical connectivity of a network. However, when evaluating the performance of transportation networks, the impact of traffic flow should also be taken into account [37; 38]. Therefore, this study proposes an URT service efficiency index weighted by traffic flow. The service efficiency between nodes \(i\) and \(j\) is defined as the weighted sum of passenger flows of all lines passing through these two nodes, divided by their shortest topological distance. \[R_{S,ij}=\sum\limits_{l\in L}(\delta_{il}+\delta_{jl})\frac{f_{l}}{d_{ij}}, \tag{2}\]  
where \(R_{S,ij}\) represents the service efficiency between nodes \(i\) and \(j\) in the URT network; \(L\) represents the line set; \(f_{l}\) represents the average daily ridership on line \(l\); \(\delta_{il}\) is a dummy variable. If node \(i\) is on line \(l\), then \(\delta_{il}=1\), otherwise, \(\delta_{il}=0\). The same applies to \(\delta_{jl}\). \begin{table}
\begin{tabular}{c c c} \hline
**Index** & **Definition** & **Formula** \\ \hline \multirow{3}{*}{Average node degree} & The average degree of all nodes in the & \\  & network (the number of connections each & \(\overline{k}=\sum\limits_{i}^{N}k_{i}/N\), \(k_{i}=\sum\limits_{j\in V_{j}\
eq i}e_{ij}\) \\  & node has with other nodes) & \\ \cline{1-1}  & The average level of clustering among & \\  & nodes in the network & \\ \cline{1-1}  & Average shortest path length & \\ \cline{1-1}  & The average shortest path length & \\ \cline{1-1}  & any two nodes in the network & \(d=\frac{2}{N(N-1)}\sum\limits_{i,j\in V_{j}\
eq j}d_{ij}\) \\ \hline \multirow{3}{*}{Closeness centrality} & Note: \(k_{i}\) is the degree of node \(i\); \(C_{i}\) is the clustering coefficient of node \(i\); \(e_{i}\) is the number of adjacent nodes of node \(i\); and \(d_{ij}\) denotes the short path distance between node \(i\) and node \(j\). \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \end{tabular}
\end{table} 
Table 1: The definitions and calculations of network topological indexes. \begin{table}
\begin{tabular}{c c c} \hline
**Index** & **Definition** & **Formula** \\ \hline \multirow{3}{*}{Degree centrality [32]} & It is the total number of the connected & \\  & edges of a node. & \\ \cline{1-1}  & It is the sum of the fractions of all-pairs & \\ \cline{1-1}  & shortest paths that pass through a node. & \(b_{q}=\sum\limits_{i=1/j=1,i\
eq j\
eq q}^{N}\frac{n_{ij}(q)}{n_{ij}}\), \(C_{B}(q)=\frac{2}{(N-1)(N-2)}b_{q}\) \\ \cline{1-1}  & It is used to measure the ability of a station & \\ \cline{1-1}  & to affect another node through the network. & \\ \cline{1-1}  & It can identify the different effects of & \\ \cline{1-1}  & neighbors of a node on it. & \\ \hline \multirow{3}{*}{Eigenvector centrality [35]} & Note: \(b_{i}\) is the betweenness centrality of node \(i\); \(n_{ij}\) is the number of shortest paths between node \(i\) and node \(j\); \(n_{ij}(q)\) is the number of shortest paths between node \(i\) and node \(j\) that pass through node \(q\); \(c_{i}\) is the closeness centrality of node \(i\); \(\lambda^{-1}\) is a proportional constant; and \(x_{j}\) is the centrality score of node \(j\). \\ \cline{1-1}  & & \\ \end{tabular}
\end{table} 
Table 2: The definitions and calculations of node centralities in complex networks.