# Measuring Method of Node Importance of Urban Rail Network Based on H Index
## 3 Method for Calculating the Importance of Urban Rail Network Nodes Based on \(h\)-Index
### Node Importance Measurement Result Based on h-Index of Urban Rail Network  
There are 234 urban rail network nodes, taking the passenger flow of the whole day as the sample, and the data collection interval is every half an hour, then the length of the passenger flow data sample is 48 (Taking full-day passenger inbound flow as an example, the calculation process of peak period and off-peak period are the same as this). After normalizing the original passenger flow matrix \(Q_{234\times 48}\), the correlation coefficient matrix between 234 stations is calculated according to formula \(r_{ij}=\frac{1}{N-1}\sum_{k=1}^{N}x_{ki}x_{kj}\):  
\[R(X)=\left[\begin{array}{cccc}1&0.972&\cdots&0.194\\ 0.972&1&\cdots&0.201\\ \vdots&\vdots&\ddots&\vdots\\ 0.194&0.201&\cdots&1\\ \end{array}\right]_{234\times 234} \tag{11}\]  
Figure 6: Node importance result based on passenger flow at urban rail station (**a**) Node importance based on passenger flow in off peak hours; (**b**) Node importance based on passenger flow in peak hours; (**c**) Node importance based on full day passenger flow.  
According to formula (1), the moment of the index coefficient between 234 stations is calculated according to formula \(h_{ij}=\left[\left(r_{ij}-minr_{ij}\right)\div\frac{1-minr_{ij}}{m}\right]\):  
\[H=\left[\begin{array}{cccc}234&227&\cdots&45\\ 227&234&\cdots&43\\ \vdots&\vdots&\ddots&\vdots\\ 45&43&\cdots&234\\ \end{array}\right]_{234\times 234} \tag{12}\]  
After obtaining the matrix of the index coefficients of the urban rail network nodes, all the nodes in the regional rail network are arranged in descending order for the index coefficients of node \(i\). When \(c_{k}^{i}\geq k\) and \(c_{k+1}^{i}<k+1\), the \(h_{r}\) index of node \(i\) is obtained. Similarly, the \(g_{r}\) index of node \(i\) can be obtained when \(\sum_{k}c_{k}^{i}\geq k^{2}\) and \(\sum_{k}c_{k+1}^{i}<\left(k+1\right)^{2}\). A graph showing the results of the \(h_{r}\) index and \(g_{r}\) index of the node is shown in Figure 7.  
The \(h_{r}\) index and \(g_{r}\) index of the nodes have similar fluctuation trends. From a vertical perspective, the \(g_{r}\) exponential curve is always above the \(h_{r}\) exponential curve, which is consistent with property 2.  
Calculate the intersection degree and intersection \(rank\) value of the top ten nodes of the two sorting sequences are:  
\[\begin{array}{c}Sim\left(top_{h_{r}}^{10},top_{g_{r}}^{10}\right)=0.7\\ rank_{h_{r}}^{10}\left(top_{h_{r}}^{10},top_{g_{r}}^{10}\right)=5.429\\ rank_{g_{r}}^{10}\left(top_{h_{r}}^{10},top_{g_{r}}^{10}\right)=4\\ \end{array} \tag{13}\]  
The first 10 elements of \(h_{r}\) and \(g_{r}\) after sorting are \(h_{r}=\left\{94,46,20,179,35,43,44,165,21,51\right\}\), \(g_{r}=\left\{179,20,44,46,35,165,21,45,164,161\right\}\), the intersection of \(h_{r}\) and \(g_{r}\) has seven elements \(\left\{179,20,44,46,35,165,21\right\}\), these seven elements are sorted in the \(h_{r}\) and \(g_{r}\) sets as (4, 3, 7, 2, 5, 8, 9) and (1, 2, 3, 4, 5, 6, 7), respectively. Then \(Sim\left(top_{h_{r}}^{10},top_{g_{r}}^{10}\right)=7/10=0.7\), \(rank_{h_{r}}^{10}(h_{r}\cap g_{r})=\left(4+3+7+2+5+8+9\right)/7=5.429\), \(rank_{g_{r}}^{10}(h_{r}\cap g_{r})=\left(1+2+3+4+5+6+7\right)/7=4\), \(\Delta rank(h_{r},g_{r},10)=rank_{h_{r}}^{10}(h_{r}\cap g_{r})-rank_{g_{r}}^{ 10}(h_{r}\cap g_{r})=1.429>0\), it means that the node identified by \(g_{r}\) index is more important than the node identified by \(h_{r}\) index.