## 2 Coupling Correlation of Complex System
### Analysis of Coupling Degree  
As the physical layer within the system, the form of coupling between the resource layer is also the most obvious: on the one hand, this coupling may result from the functional/logical coupling generated by each resource element serving the same function; on the other hand, it may also cause direct material or information transfer between resource elements, thus introducing specific coupling relationships. Both of the above two coupling forms can be defined as direct coupling. In contrast, a more complex form of indirect association between groups of coupled resource elements is generated due to the addition of the resource-sharing form. This form of coupling can be defined as indirect/cascading coupling. In order to quantitatively describe the direct and indirect coupling relationships within the system hierarchy, this study takes the resource layer as an example to define and distinguish the two coupling concepts. 1. Direct coupling degree matrix  
The direct coupling degree is used to characterize the direct coupling relationship between elements. It represents the situation where there are direct information interactions, material exchanges or being occupied by the same other layer elements in the layer. The direct coupling degree matrix \(C^{d}\) is represented in Matrix (4):  
\[C^{d}=\left(C^{d}_{ij}\right)_{n\times n}= \begin{array}{c}r_{1}\\ r_{2}\\ \vdots\\ r_{n}\end{array} \left[ \begin{array}{cccc}r_{1}&r_{2}&\ldots&r_{n}\\ \left[\begin{array}{cccc}C^{d}_{11}&C^{d}_{12}&\ldots&C^{d}_{1n}\\ C^{d}_{21}&C^{d}_{22}&&C^{d}_{2n}\\ \vdots&\ddots&\vdots\\ C^{d}_{n1}&C^{d}_{n2}&\cdots&C^{d}_{mn}\end{array}\right]\end{array} \right. \tag{4}\]  
where \(C^{d}_{ij}=0\) means no direct correlation; \(C^{d}_{ij}=1\) means correlation degree between resource element \(r_{i}\) and resource element \(r_{j}\) is 1, that, is, the fault propagation from resource element \(r_{i}\) to resource element \(r_{j}\) only needs 1 step. 2. Indirect coupling degree matrix  
According to the fault propagation theory and cascading failure theory, the fault of a single element will not only affect the element itself, but also cause a cascading effect by the correlation between elements, causing the fault propagation and diffusion, and the more serious situation may affect the normal operation of the whole system. Thus, simply establishing the concept of direct coupling degree is insufficient to assess the potential risk introduced by multiple coupling correlation of elements. In contrast, the indirect coupling degree is more efficient to reflect the degree of such risk. The indirect coupling degree matrix \(C^{c}_{ij}\) characterizes the indirect coupling relationships between elements, which is an extension of the direct coupling degree. It can be represented by the indirect coupling degree matrix \(C^{c}\) as presented in Matrix (5). \[C^{c}=\left(C^{c}_{ij}\right)_{n\times n}= \begin{array}{c}r_{1}\\ r_{2}\\ \vdots\\ r_{n}\end{array} \left[ \begin{array}{cccc}r_{1}&r_{2}&\ldots&r_{n}\\ \left[\begin{array}{cccc}C^{c}_{11}&C^{c}_{12}&\ldots&C^{c}_{1n}\\ C^{c}_{21}&C^{c}_{22}&&C^{c}_{2n}\\ \vdots&\ddots&\vdots\\ C^{c}_{n1}&C^{c}_{n2}&\cdots&C^{c}_{mn}\end{array}\right]\end{array} \right. \tag{5}\]  
Based on the direct coupling degree matrix \(C^{d}\), the shortest fault propagation path order is calculated based on the Floyd algorithm, and that means the indirect coupling degree matrix \(C^{c}\) is generated. Among them, the element \(C^{c}_{ij}\) in the matrix is a natural number. When \(C^{c}_{ij}=0\), it means no indirect coupling relationship. When \(C^{c}_{ij}=n\), it means that coupling correlation degree between resource element \(r_{i}\) and resource element \(r_{j}\) is n, indicating that fault propagation from resource element \(r_{i}\) to resource element \(r_{j}\) needs n steps. The basic process of the Floyd algorithm is to start from the direct coupling matrix \(C^{d}\) and recursively update n times.