# Human-Related Hazardous Events Assessment for
## 2 Materials and Methods
### Calculation for the Node Criticality  
#### 2.3.1 Basic Criticality Calculation for Nodes within CN  
The topology of the CN can be described by the parameters associated with degree and weight. It is noticeable that the weight-related parameter is equal to 1 in the case of a single accident report. To implement the topological description for the developed directed CN, the network is represented as:  
\[G=(V,E) \tag{1}\]  
where \(V=\{v_{1},v_{2},...,v_{N}\}\) and \(E=\{e_{1},e_{2},...,e_{M}\}\) represent the set of nodes and the set of edges, respectively. In the present study, \(v_{i}\) indicates the identified hazardous event, while \(e_{i}\) denotes the logic relationship. Then, an adjacency matrix \(\Lambda\) is employed to quantitatively describe \(G=(V,E)\), and the elements in the adjacency matrix are determined by:  
\[A_{ij}=\left\{ \begin{array}{cc}a_{ij}\cdot w_{ij}&i\to j\\ 0&else\end{array} \right. \tag{2}\]  
where \(a_{ij}(i,j\in V)\) is equal to 1 when \(v_{j}\) is triggered by \(v_{i}\), and is zero otherwise. The weight between the node \(v_{i}\) and the node \(v_{j}\) is expressed by \(w_{ij}\). The meaning of weight is to make the connection strength between the nodes are quantified. The weight is divided into two kinds: in-weight and out-weight. In general, the characteristics of the CN are obtained by node strength and cross-weight and can be determined by the correlation calculation of the weight. \[T_{\text{w}}(i)= S_{\text{in}}(i)+S_{\text{out}}(i)= \sum_{j\in V}w_{ji}+\sum_{j\in V}w_{ij} \tag{3}\] \[C_{\text{w}}(i)= S_{\text{in}}(i)\cdot S_{\text{out}}(i)= \sum_{j\in V}w_{ji}\cdot\sum_{j\in V}w_{ij} \tag{4}\]  
Figure 2: CN obtained from the accident report. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Expert** & **Age** & **Occupation** & **Educational Level** & **Certificate Rank** & **Job Tenure** \\ \hline Expert & 53 & Professor in & Doctor of Naval & & He is a professor employed in a maritime university and & \\  & & university & Architecture and & & his research interests involve operational safety aspects & \\  & & & Marine Engineering & & in shipbuilding & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Another important parameter for a typical, directed CN refers to the degree of node, which is defined as the sum of nodes connecting or being connected to the objective node. In the present study, the degree of the objective node is denoted by \(d(i)\). Similarly, the degree is also divided as in-degree and out-degree, which can be determined, respectively, by:  
\[d_{\text{in}}(i)=\sum\limits_{j\in V}a_{ji} \tag{5}\]  
\[d_{\text{out}}(i)=\sum\limits_{j\in V}a_{ij} \tag{6}\]  
According to the equations expressed in (5) and (6), the in-degree of a node is the total number of nodes affecting the objective node, and the out-degree of a node refers to the total number of nodes affected by the objective node. Then, the total degree of the objective node can be obtained by:  
\[T_{\text{d}}(i)=d_{\text{in}}(i)+d_{\text{out}}(i)=\sum\limits_{j\in V}a_{ji}+ \sum\limits_{j\in V}a_{ij} \tag{7}\]  
The total degree is generally used to evaluate the activeness of nodes within a network according to graph theory; however, the difference between nodes characterized by the same total degree cannot be distinguished from the perspective of activeness. Therefore, the concept of the cross-degree is introduced in this study, which is mainly applied to distinguish nodes with the same total degree:  
\[C_{\text{d}}(i)=d_{\text{in}}(i)\cdot d_{\text{out}}(i)=\sum\limits_{j\in V}a_ {ji}\cdot\sum\limits_{j\in V}a_{ij} \tag{8}\]  
In the present study, node criticality was determined by the improved K-shell decomposition algorithm, which is implemented with the precondition of calculating the basic criticality as the input for the improved K-shell decomposition algorithm.