## 2 Methodology
### Evaluation of Hazardous Factors with Complex Network
#### 2.2.2 Evaluation Based on Topological Indexes  
The HFN depicts the topological structure of hazardous factors from the perspective of causal relationships. It provides a model foundation for evaluation. By the use of topological indexes, the evaluation of hazardous factors can be performed based on the HFN. Nevertheless, the existing indexes [32; 33; 35], such as degree, clustering coefficient, and betweenness, are designed for homogeneous networks or heterogeneous networks consisting of two types of nodes. The HFN proposed in this paper is a kind of heterogeneous network including three types of nodes, i.e., causal factor nodes, unsafe control action nodes, and hazard nodes. To evaluate the hazardous factors by a topological analysis, some topological indexes should be customized in accordance with the structural features of the HFN. A topological index known as causal connection density of the HFN is proposed to measure the overall connection feature of the network, as shown in Equation (4). It reflects the causal connection degree of the hazardous factors in the HFN. This index is calculated by the ratio of the number of actual edges to the number of possible edges, where the product of \(N_{CF}\) and \(N_{UC}\) is the maximum possible number of cause-effect edges between causal factors and unsafe control actions, the product of \(N_{UC}\) and \(N_{H}\) is the maximum possible number of cause-effect edges between unsafe control actions and hazards, and half of the product of \(N_{CF}\) and \(N_{CF}\) is the maximum possible number of cause-effect edges among causal factors. \[CCD_{HFN}=\sum\limits_{All\ nodes}(AM_{ij})/(N_{CF}\cdot N_{UC}+N_{UC}\cdot N_{ H}+N_{CF}\cdot N_{CF}/2) \tag{4}\]  
To evaluate the overall causal path density of the HFN, the index of path density is proposed and calculated by Equation (5). As shown by Equation (5), the product of \(N_{CF}\) and \(N_{H}\) is the maximum possible number of causal paths between causal factors and hazards. The path density denotes the scale of actual causal paths from causal factors to hazards. In other words, the higher the value of this index, the higher the number of causal paths, and hence the more likely the hazards are to be caused. \[PD_{HFN}=\sum\limits_{ \begin{array}{ll}i\in V_{CF},\\ h\in V_{H}\end{array} }(RM_{ii})/(N_{CF}\cdot N_{H}) \tag{5}\]  
As mentioned, the hazardous factor nodes in the HFN include causal factor nodes and unsafe control action nodes. These hazardous factor nodes and their causal relationships constitute hazard causal paths. To further explore the degree of causal correlation between nodes, two indexes of causal correlation are designed. One is the index of active correlation in Equation (6). It measures the degree of causal correlation between a hazardous factor node \(i\) and the hazardous factor nodes caused by node \(i\). As shown by Equation (6), the causal correlation is inversely proportional to the shortest path lengths, i.e., the longer the path, the weaker the causal correlation. Similarly, another index, known as passive correlation, is designed in Equation (7). It denotes the degree of causal correlation between a hazardous factor node \(i\) and the causal factors nodes that can cause node \(i\). \[AC_{i}=\sum_{j\in V_{CF},V_{ULC}}RM_{ij}/\sum_{j\in V_{CF},V_{ULC}}SPM_{ij} \tag{6}\]  
\[PC_{i}=\sum_{j\in V_{CF}}RM_{ji}/\sum_{j\in V_{CF}}SPM_{ji} \tag{7}\]  
If a hazardous factor node appears on several different hazard causal paths, it is a key node for blocking these causal paths. It is significant to identify this kind of node for hazard control. In view of this, an index derived from the traditional betweenness index is proposed in accordance with the structural features of the HFN, and is shown in Equation (8). The index reveals the degree of intermediary for a hazardous factor in hazard causal paths, i.e., the number of paths through a causal factor node or an unsafe control action node. The higher the value of the index of a hazardous factor node is, the higher the degree of intermediary for it is.