## 2 Methodology
### Evaluation of Hazardous Factors with Complex Network  
#### 2.2.1 Construction of Causation Network  
To evaluate the hazardous factors identified by STPA, a network of hazardous factors and their causal relationships must be constructed first. In such a network, the hazardous factors, including both unsafe control actions and their causal factors, are abstracted as nodes. The cause-effect links between hazardous factors are represented by directed edges connecting the nodes. Besides, hazards and their corresponding causal links with unsafe control actions are also represented by nodes and edges. The above network is known as the hazardous factor network (HFN). Its definition is shown, as follows. **Definition 1**. _The HFN is a quadruple \(<\)V\({}_{\text{CS}}\), V\({}_{\text{UC}}\), V\({}_{H}\), R\(>\), where_  
1. \(V_{\text{CF}}\) _represents the set of causal factor nodes, i.e., the causal factors leading to unsafe control actions. The number of the causal factor nodes is denoted as N\({}_{\text{CF}}\)._
2. \(V_{\text{UC}}\) _represents the set of unsafe control action nodes, i.e., the unsafe control actions resulting in hazards. The number of nodes in V\({}_{\text{UC}}\) is denoted as N\({}_{\text{UC}}\)._
3. \(V_{H}\) _refers to the set of hazard nodes, the number of which is denoted as N\({}_{H}\)._
4. _R= ((V\({}_{\text{CF}}\), V\({}_{\text{UC}}\)), (V\({}_{\text{UC}}\), V\({}_{H}\))) is the set of cause-effect relationships between nodes, and exists in the form of directed edges. In R, (V\({}_{\text{CF}}\), V\({}_{\text{UC}}\)) refers to the cause-effect links between causal factor nodes and unsafe control action nodes. Similarly, (V\({}_{\text{UC}}\), V\({}_{H}\)) refers to the cause-effect links between unsafe control action nodes and hazard nodes._  
By Definition 1, the HFN can be constructed through the adjacency matrix defined in Equation (1). \[AM_{ij}=\left\{ \begin{array}{l}1\,\ (i,\ j)\in R\\ 0\,\ (i,\ j)\
otin R\end{array} \right. \tag{1}\]  
where both \(i\) and \(j\) represent the nodes in \(\{V_{\text{CF}}\), \(V_{\text{UC}}\), \(V_{\text{H}}\}\), and \((i,j)\) represents the relationships between the two nodes. If there is a cause-effect link between \(i\) and \(j\), the entry of adjacency matrix is equal to 1, i.e., there is a directed edge connecting the nodes \(i\) and \(j\). It is worth noting that the entry \(AM_{ij}\) with the value of 1 means the HFN is a kind of unweighted network. This is because the hazardous factors and their cause-effect links are identified by the qualitative STPA, i.e., it is difficult to capture the quantification of the strength of the cause-effect relationship by STPA. In fact, the quantified values of cause-effect relationships change with different system implementation and complex operational environments. It is difficult to obtain the quantified cause-effect strength. Without loss of generality, the cause-effect relationships are represented by unweighted edges in this paper. To facilitate the subsequent evaluation, two kinds of structural matrixes are introduced based on the definition of the adjacency matrix. One is the shortest path matrix,which reflects the features of causal distances among hazardous factors and is defined by Equation (2). \[SPM_{ij}=\left\{ \begin{array}{ll}Min\{\ \sum\limits_{m,n\in Pk}AM_{mn}\},&if\ P \
eq O\\ \infty,&otherwise\end{array} \right. \tag{2}\]  
where both \(i\) and j represent the nodes in \(\{V_{CF},\ V_{UC},\ V_{H}\}\), \(P=\{Pk\ |\ k=1,\ 2,\ \ldots,\ n\}\), \(Pk\) is a causal path from the node \(i\) to the node \(j\), and m and \(n\) are the nodes on the path \(Pk\). If there are causal paths between \(i\) and \(j\), the value of the entry \(SPM_{ij}\) is the minimum of the path lengths from \(i\) to \(j\). This matrix can be calculated by [PERSON]'s algorithm [42]. Another matrix is the reachability matrix, which depicts the indirect causal relationships between hazardous factors and can be calculated by Equation (3). \[RM_{ij}=\left\{ \begin{array}{ll}0,&if\ SPM_{ij}=\ \infty\\ 1,&otherwise\end{array} \right.