## 3 Results
### Improved GRE-TOPSIS Method
#### 3.2.3 Construct the Weighted Evaluation Indicator Matrix  
To accurately calculate the importance evaluation value of the nodes, this paper calculates weights using both the gray correlation and entropy weight methods. The entropyweight method objectively determines the weight based on the amount of information difference between the indicators. The gray correlation method, as an evaluation method measuring the similarity or difference in development trends between factors, can reflect the changes in the indicators contained within the evaluation scheme and can comprehensively evaluate multi-indicator problems.  
First, \(U=\left(u_{ij}\right)_{m\times n}\) is normalize to obtain the sample data value of the \(i\)-th node under the \(j\)-th evaluation indicator as a proportion of that indicator as:  
\[P_{ij}=\frac{u_{ij}}{\sum\limits_{i=1}^{m}u_{ij}}, \tag{9}\]  
where \(0\leq P_{ij}\leq 1\), \(\sum\limits_{i=1}^{m}P_{ij}=1\). Then the entropy of each evaluation metric is calculated by column, then the entropy value of the \(j\)-th evaluation metric of the node is formulated as follows:  
\[E_{j}=-k\sum\limits_{i=1}^{m}P_{ij}\ln P_{ij}, \tag{10}\]  
where \(0\leq E_{j}\leq 1\), \(k=\frac{1}{\ln m}\). The information redundancy of the \(j-\)th evaluation indicator is calculated as \(R_{j}=1-E_{j}\). The greater the value of redundancy, the greater the impact of the evaluation metric on the evaluation of node importance and the greater the weight attached to it. Thus, the first part of the weights is obtained, and the entropy weight of the \(j-\)th evaluation indicator is calculated as follows:  
\[\alpha_{j}=\frac{R_{j}}{\sum\limits_{j=1}^{n}R_{j}} \tag{11}\]  
where \(\sum\limits_{j=1}^{n}\alpha_{j}=1\).  
Moreover, for the calculation of the gray correlation weight, the largest element of each row of the matrix \(U=\left(u_{ij}\right)_{m\times n}\) is selected as the reference object \(Z\):  
\[Z=\begin{bmatrix}z_{1}\\ z_{2}\\ \vdots\\ z_{m}\end{bmatrix} \tag{12}\]  
where \(z_{i}=\max\left(u_{ij}\right),i=1,2,\cdots,m,j=1,2,\cdots,n\). Then, the gray correlation coefficient \(\xi_{ij}\) between \(u_{ij}\) and the reference object \(z_{i}\) is calculated as follows:  
\[\xi_{ij}=\frac{\min\limits_{i=1,2,\cdots,m}\min\limits_{j=1,2,\cdots,n}\left| z_{i}-u_{ij}\right|+\rho\max\limits_{i=1,2,\cdots,m}\max\limits_{j=1,2, \cdots,n}\left|z_{i}-u_{ij}\right|}{\left|z_{i}-u_{ij}\right|+\rho\max\limits _{i=1,2,\cdots,m}\max\limits_{j=1,2,\cdots,n}\left|z_{i}-u_{ij}\right|} \tag{13}\]  
where \(\rho\) is the resolution factor and is usually set as 0.5.  
Thus, the gray correlation weight is obtained as \(\beta_{j}=\sum\limits_{i=1}^{m}\xi_{ij},j=1,2,\cdots,n\). Combining the two methods to calculate the weight makes the calculation results more objective. Determine the weight of each indicator as \(\omega_{j}=a\alpha_{j}+b\beta_{j}\), \(j=1,2,\cdots,n,a+b=1\), then \(\omega=\left(\omega_{1},\omega_{2},\cdots,\omega_{n}\right)\). Weight the matrix \(U\) to obtain the weighted matrix \(Q=\left(q_{ij}\right)_{m\times n}\), as follows:  
\[Q=\left(\omega_{j}u_{ij}\right)_{m\times n}=\begin{pmatrix}\omega_{1}u_{11}& \cdots&\omega_{n}u_{1n}\\ \vdots&\ddots&\vdots\\ \omega_{1}u_{m1}&\cdots&\omega_{n}u_{mn}\end{pmatrix}=\begin{pmatrix}q_{11}& \cdots&q_{1n}\\ \vdots&\ddots&\vdots\\ q_{m1}&\cdots&q_{mn}\end{pmatrix}. \tag{14}\]