## 4 Case Analysis
### Construction of Node Importance Evaluation System  
In traditional node identification research, researchers usually use node betweenness, node degree, clustering coefficient, and node centrality as topology structure or vulnerability indicators for importance analysis and ranking. In the complex network \(G\) studied in this article, the indicator data distributed by nodes is shown in Figure 4.  
Next, the decision matrix is constructed according to the calculated values of the indicators at each node, and the indicators in the matrix are normalized to obtain the normalized decision matrix. The calculation results of the relevant indicators are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Node** & **Degree** & **Betweenness** & **Clustering Coefficient** & **Closeness Centrality** \\ \hline
**0** & Baotou & 3 & 0.056518598 & 0 & 0.177350427 \\
**1** & Hengshui & 4 & 0.035033769 & 0 & 0.228021978 \\
**2** & Yangquan & 2 & 0.063446958 & 0 & 0.22675956 \\
**3** & Jingfeng Port & 5 & 0.00753872 & 0.4 & 0.238505747 \\
**4** & Beijing & 7 & 0.357347351 & 0.047619048 & 0.284246575 \\
**5** & Shuozhou & 2 & 0.039244655 & 0 & 0.20906801 \\
**6** & Shanhaiguan & 2 & 0.079075885 & 0 & 0.213367609 \\
**7** & Liaoheng & 4 & 0.043588158 & 0 & 0.22916898 \\
**8** & Ankang & 1 & 0 & 0.163064833 \\
**9** & Chifeng & 4 & 0.205155118 & 0 & 0.24702381 \\
**10** & Manzhouli & 1 & 0 & 0.133446154 \\
**11** & Nanjing & 5 & 0.049207013 & 0.2 & 0.239884393 \\
**12** & Jiaozuo & 3 & 0.028218839 & 0 & 0.205445545 \\
**13** & Handan & 4 & 0.105028972 & 0 & 0.24340176 \\
**14** & Wuhan & 4 & 0.038749177 & 0.166666667 & 0.220159151 \\
**82** & Tangshan & 3 & 0.024198284 & 0 & 0.249249249 \\
**83** & Hohhot & 2 & 0.052658149 & 0 & 0.189066059 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Table of normalized decision matrix calculation results (partial).  
Figure 3: Topology diagram of coal rail-water transportation network. (Using Gephi v0.9.2 with the latitude and longitude from Google Maps).  
To mitigate the potential weight error when using a single model for evaluation, this paper adopts two different analytical methods: the entropy weighting method and the gray relational degree method. We determine the weight value for each indicator and then average the weights of the two methods to achieve a comprehensive assessment of node importance. The calculated weights are shown in Table 2.  
The rankings and weights of the indicators are as follows: (1) Clustering Coefficient, with a weight of 0.4198; (2) Betweenness, with a weight of 0.2035; (3) Closeness Centrality, with a weight of 0.1903; (4) Degree, with a weight of 0.1865.  
By inputting the metric weights into the corresponding decision matrix, we can calculate the importance values for each node in the basic network, as shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Entropy Weight Method** & **Grey Relational Degree** & **Average Weight** \\ \hline
**Degree** & 0.111596374 & 0.261344274 & 0.186470324 \\
**Betweenness** & 0.202804217 & 0.204224974 & 0.203514595 \\
**Clustering Coefficient** & 0.635896556 & 0.203578962 & 0.419737759 \\
**Closeness Centrality** & 0.049702853 & 0.330851791 & 0.190277322 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Table of weight calculation results.  
Figure 4: Topology diagram of coal rail-water transportation network: (**a**) Degree; (**b**) Betweenness; (**c**) Clustering Coefficient; (**d**) Closeness Centrality.