## 4 Case Study and Discussion
### Coupling Correlation of Complex System
#### 4.1.2 Coupling Degree and Related Factors  
1. Direct coupling degree matrix  
Direct coupling relationship caused by different resource elements serving same function can be presented by direct coupling degree matrix \(C^{d}\) shown in Matrix (19). \[C^{d}=\left(C^{d}_{ij}\right)_{9\times 9}=\left[ \begin{array}{ccccccccc}0&1&1&1&0& 0&0&0&0\\ 1&0&1&1&0&0&0&0&0\\ 1&1&0&1&1&1&0&1&0\\ 1&1&1&0&1&1&0&1&0\\ 0&0&1&1&0&1&1&1\\ 0&0&0&0&1&1&0&1&1\\ 0&0&1&1&1&1&0&1\\ 0&0&0&0&1&1&1&1&0\end{array} \right] \tag{19}\]  
2. Indirect coupling degree  
Based on Floyd algorithm, fault propagation path order is calculated referring to direct coupling matrix \(C^{d}\), and then, the indirect coupling degree matrix \(C^{c}\) and the shortest path matrix \(C^{r}\) of fault propagation are generated as shown in Matrix (20) and Matrix (21), respectively. \(C^{r}_{ij}\) represents the next element that the fault propagation from element \(i\) to element \(j\) should pass. For instance, fault propagation from element 1 to element 9, according to \(C^{r}_{19}=3\), it can be inferred that fault in element 1 will propagate to element 3 first. Then, according to \(C^{r}_{39}=5\), it can be inferred that fault will propagate from element 3 to element 5 second. Finally, according to \(C^{r}_{59}=9\), it can be seen that fault will propagate from element 5 to element 9. Therefore, the fault propagation path from element 1 to element 9 is formed: \(1\to 3\to 5\to 9\). So, other fault propagation paths can be deduced by analogy. \[C^{c}=\left(C^{c}_{ij}\right)_{9\times 9}=\left[ \begin{array}{ccccccccc}0&1&1&1&2&2& 3&2&3\\ 1&0&1&1&2&2&3&2&3\\ 1&1&0&1&1&1&2&1&2\\ 1&1&1&0&1&1&2&1&2\\ 2&2&1&1&0&1&1&1\\ 2&2&1&1&1&0&1&1&1\\ 3&3&2&2&1&1&0&1&1\\ 2&2&1&1&1&1&0&1\\ 3&3&2&2&1&1&1&1&0\end{array} \right] \tag{20}\]\[C^{r}=\left(C^{r}_{ij}\right)_{9\times 9}=\left[ \begin{array}{cccccccc}1&2&3&4&3&3&3&3&3 \\ 1&2&3&4&3&3&3&3&3\\ 1&2&3&4&5&6&5&8&5\\ 1&2&3&4&5&6&5&8&5\\ 3&3&3&4&5&6&7&8&9\\ 3&3&3&4&5&6&7&8&9\\ 5&5&5&5&5&6&7&8&9\\ 3&3&3&4&5&6&7&8&9\\ 5&5&5&5&5&6&7&8&9\\ 3&3&3&4&5&6&7&8&9\\ 5&5&5&5&5&6&7&8&9\\ 5&5&5&5&5&6&7&8&9\\ \end{array} \right] \tag{21}\]  
3. Potential severity  
The potential severity effect is greatest when there is a direct coupling correlation relationship between the elements (the coupling degree is 1), so the corresponding potential severity value is set to 1. In addition, when the coupling degree is 5 or above, the degree of effect is the smallest, and the corresponding potential severity value is set to 0.1. Based on Equation (7), \(S_{p}(1)=\frac{1}{1+a\left(1-c\right)^{2}}=1\), \(S_{p}(5)=\frac{1}{1+a\left(5-c\right)^{2}}=0.1\), as shown in equation set (22), and then solving equation set (22), obtaining \(c=1\), \(a=0.56\). Then, based on Equation (7), obtaining \(S_{p}(2)=0.64\),\(S_{p}(3)=0.31\),\(S_{p}(4)=0.17\). Therefore, potential severity matrix \(S_{p}\) is shown as Matrix (23). \[S=\left(S_{p}\right)_{9\times 9}=\left[ \begin{array}{cccccccc}0.00&1.00&1.00& 1.00&0.64&0.64&0.31&0.64&0.31\\ 1.00&0.00&1.00&0.64&0.64&0.31&0.64&0.31\\ 1.00&1.00&0.00&1.00&1.00&0.64&1.00&0.64\\ 1.00&1.00&1.00&0.00&1.00&0.64&1.00&0.64\\ 0.64&0.64&1.00&1.00&0.00&1.00&1.00&1.00\\ 0.64&0.64&1.00&1.00&0.00&1.00&1.00&1.00\\ 0.64&0.64&1.00&1.00&1.00&0.00&1.00\\ 0.64&0.64&1.00&1.00&1.00&0.00&1.00\\ 0.64&0.64&1.00&1.00&1.00&1.00&0.00\\ \end{array} \right] \tag{23}\]  
In addition, considering the safety critical grade of the resource elements \(SCG_{1\times 9}=\left\{1,\,2,\,3,\,3,\,2,\,1,\,2,\,2,\,3\right\}\), and based on Equation (7), the final potential severity matrix \(S\) is presented as Matrix (24). \[S=S_{9\times 9}=\left[ \begin{array}{cccccccc}0.00&2.00&3.00&3.00&1.28&0.64 &0.62&1.28&0.93\\ 1.00&0.00&3.00&3.00&1.28&0.64&0.62&1.28&0.93\\ 1.00&2.00&0.00&3.00&2.00&1.00&1.28&2.00&1.92\\ 1.00&2.00&3.00&0.00&2.00&1.00&1.28&2.00&1.92\\ 0.64&1.28&3.00&3.00&0.00&1.00&2.00&2.00&3.00\\ 0.31&0.62&1.92&1.92&2.00&1.00&2.00&2.00&0.00\\ \end{array} \right] \tag{24}\]  
in the direct propagation time \(C_{d}^{T}\).