# Measuring Method of Node Importance of Urban Rail Network Based on H Index
## 2 Preliminaries
### Node Importance Measurement Index Based on PageRank Algorithm  
Google's PageRank is a variant of the centrality measure of feature vectors. Google uses the PageRank algorithm to compute the PageRank value of each page, and then ranks the pages in the result set. The higher the PageRank value of a page, the higher the ranking of the page in the result set.  
Some scholars have applied PageRank algorithm to the research of network node importance [10; 11]. Let undirected graph \(G=\left(V,E\right)\), whose adjacency matrix is \(A=\left(e_{ij}\right)_{i,j=1}^{N}\in R^{N\times N}\). If there is an edge between node \(i\) and node \(j\), then \(e_{ij}=1\), and vice versa \(e_{ij}=0\). The transition matrix  
\begin{table}
\begin{tabular}{c p{142.3 pt} p{142.3 pt}} \hline \hline
**Index** & **Definition** & **Calculation Method** \\ \hline \multirow{4}{*}{Degree Centrality} & Degree centrality (DC) is the most direct measure of the centrality of a node. The greater the degree of a node, the higher the degree of centrality of the node, and the more important the node is in the network. & \(C_{D}(i)=\frac{\sum_{i\in\left(i\right)}x_{ij}}{N-1},N\) is the total number of network nodes; \(\Gamma(i)\) represents a collection of neighbor nodes of node \(i\) \\ \hline \multirow{4}{*}{Betweenness Centrality} & The Betweenness centrality (BC) indicator is used to measure the importance of the individual status of a social network. The measure of BC is the shortest number of paths through a node. & \(C_{B}(i)=\sum_{j\in\left(i\right)}\frac{\sum_{i\in\left(i\right)}x_{ij}}{S_{d} }/C_{N-1}^{2},N\) is the total number of nodes in the network; \(S_{d}\) represents the number of all shortest paths through node \(i\) in the shortest path from node \(s\) to node \(t\). \\ \hline \multirow{4}{*}{Closeness Centrality} & Closeness centrality (CC) is used to measure the ability of a node to influence other nodes through the network. Its metric is the topological distance between nodes. & \(C_{C}(i)=\left[\sum_{j}^{N}d_{ij}\right]^{-1}\), \(d_{ij}\) represents the shortest distance between node \(i\) and node \(j\). \\ \hline \multirow{4}{*}{Eigenvector Centrality} & Eigenvector centrality (EC) is an indicator that takes into account the number of neighbor nodes and the importance of neighbor nodes. Its basic idea is that the neighbor nodes of the node have different effects on the node. & \(C_{E}(i)=\frac{1}{\lambda}\sum_{j\in\left(i\right)}^{N}a_{ij}x_{j}\), \(\Gamma(i)\) is the set of neighbor nodes of node \(i\), where \(\lambda\) is a constant. \\ \cline{1-1}  & & \(C_{E}(i)=\frac{1}{\lambda}\sum_{j\in\left(i\right)}^{N}a_{ij}x_{j}\), \(\Gamma(i)\) is the set of neighbor nodes of node \(i\), where \(\lambda\) is a constant.  
is represented as \(A^{*}=\left(a_{ij}\right)_{i,j=1}^{N}\), among them, \(a_{ij}\) is the reciprocal of the sum of all the elements of the corresponding column in the adjacency matrix \(A\), as shown in formula (1).  
\[a_{ij}=\begin{cases}\frac{1}{\sum_{i=1}^{N}c_{ij}}&e_{ij}=1\\ 0&e_{ij}=0\end{cases}\leq i,j\leq N \tag{1}\]  
Then the PageRank algorithm can be expressed as:  
\[G=(1-\alpha)A^{*}+\frac{\alpha}{N}I_{N\times N} \tag{2}\]  
Among them, \(\alpha\) represents the damping factor, \(0<\alpha<1\), usually \(\alpha=0.15\); \(I_{N\times N}\) is the unit matrix.  
Let the maximum eigenvalue of matrix \(G\) be \(\lambda_{1}=1\) and the corresponding eigenvector be \(X_{1}\). Using Perron-Frobenius theorem [17] to solve \(X_{1}\):\(G\):\(X=1\):\(X_{1}\). If the solution result is \(X_{1}=x_{1}(1),x_{1}(2),\cdots,x_{1}(i),\cdots,x_{1}(N)\), then \(x_{1}(i)\) is the importance value of node \(i\).