# Study on Early Warning Methods for Shipping Input Risks Under Consideration of Public Health Events
## 5 Shipping Network Node Ranking Based on Node Importance  
Considering the availability of actual data, this paper collects data from the aviation passenger layer and maritime bulk cargo layer for calculations. Node importance metrics can help determine the importance of nodes within the entire network. Using clustering coefficient, betweenness centrality, closeness centrality, degree centrality, and eigenvector centrality as metrics for evaluating node importance, we rank the importance of shipping network nodes based on the TOPSIS model modified by a weight extraction method. The main steps are as follows:  
Step 1: Based on the metric data \(a_{ij}(i=1,2,\ldots,m;j=1,2,\ldots,n)\), where \(i\) is the number of evaluation targets and \(j\) is the number of metrics, establish the original evaluation metric system matrix \(A_{nm}\):  
\[A_{mn}= \begin{bmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{m1}&\cdots&a_{mn}\end{bmatrix} \]  
Check whether there are negative numbers in the input matrix. If negative numbers are present, standardize them to a non-negative range, and denote the standardized matrix as \(\widetilde{Z}_{ij}\). The calculation formula is as follows:  
\[\widetilde{Z}_{ij}=\frac{a_{ij}}{\sqrt{\sum_{i}^{n}a_{ij}^{2}}} \tag{15}\]Step 2: Calculate the information entropy of each index and compute the information utility value. Normalize the results to obtain the weight for each index. The formula for calculating information entropy is as follows:  
\[e_{j}=-(Ln(m))^{-1}{\sum_{j=1}^{m}P_{ij}}\ln(P_{ij}) \tag{16}\]  
Step 3: Compute the weight \(W_{j}\) based on the information extracted from each index to construct a weighted normalized matrix \(R_{mm}\). \[W_{j}=\frac{1-e_{j}}{n-\sum_{j=1}^{n}e_{j}},R_{mm}=\left\{r_{ij}\right\}_{m \times n}=W_{j}\times\widetilde{Z}_{ij} \tag{17}\]  
Step 4: Calculate the maximum value \(X^{+}\) and the minimum value \(X^{-}\), where \(X^{+}\) is the maximum value for each evaluation index, and \(X^{-}\) is the minimum value for each evaluation index. \[X^{+}=\left\{r_{1}^{+},r_{2}^{+},\ldots,r_{n}^{+}\right\}=(\max(r_{11},r_{21},\ldots,r_{n1}),\max(r_{12},r_{22},\ldots,r_{n2}),\ldots,\max(r_{1m},r_{2m}, \ldots,r_{nm})) \tag{18}\]  
\[X^{-}=\left\{r_{1}^{-},r_{2}^{-},\ldots,r_{n}^{-}\right\}=(\min(r_{11},r_{21},\ldots,r_{n1}),\min(r_{12},r_{22},\ldots,r_{n2}),\ldots,\min(r_{1m},r_{2m}, \ldots,r_{nm})) \tag{19}\]  
Step 5: Calculate the closeness \(S_{i}\) of each node to the maximum and minimum values, where \(0<S_{i}<1\). Each node is sorted according to the magnitude of the closeness value, with a larger value indicating that the node is more important. \[D_{i}^{+}=\sqrt{\sum_{j=1}^{n}\left(r_{ij}-r_{j}^{+}\right)^{2}},D_{i}^{-}= \sqrt{\sum_{j=1}^{n}\left(r_{ij}-r_{j}^{-}\right)^{2}},\;S_{i}=\frac{D_{i}^{- }}{D_{i}^{-}+D_{i}^{+}} \tag{20}\]  
Step 6: Calculate contribution \(\omega\). Step 7: Calculate node mutual coupling importance. \[\sum_{ijn}(\omega_{1i}*\overline{q_{1 in}}+\omega_{2j}*\overline{q_{2 jn}}),i= 1,2,j=1,2,3,4,n=1,2,\ldots,241 \tag{21}\]  
Based on the above steps 1-5, calculate the importance of nodes in the aviation and maritime networks. Then, according to steps 6-7, calculate the node mutual coupling importance in the global aviation and maritime mutual coupling network. The importance indicators of some nodes and the results of node importance are shown in Tables 2-6 below: