## 4 Case Study and Discussion
### Discussion  
1. According to the sorting results in Figure 6, Figure 7, and Pareto Rule, elements 5 and 6 can be defined as the safety-critical elements; and elements 1, 2, 3, 4, 7, 8, and 9 are defined as safety-general elements.
2. Based on Figure 8, element 5 and 6 are in the serious risk region; elements 3, 4, 7, 8, and 9 are in the ALARP region; and elements 1 and 2 are in the negligible region.
3. From Figure 9, elements 1 and 2 have higher uncertainty; elements 6, 7, and 9 have moderate uncertainty; and elements 3, 4, 5, 8 are with lower uncertainty.
4. In summary, elements 5 and 6 are the safety-critical elements and located in a serious risk region, which has a serious effect on the overall system. Simultaneously, it has a certain degree of uncertainty. Therefore, corresponding measures must be taken to ensure the safety of elements 5 and 6 in order to decrease the system risk. In other words, more attention must be paid to Signal Processing Module in this avionics system. Elements 3, 4, 7, 8, and 9 are in the ALARP region and have lower uncertainty. This means that the risk caused by these elements in the region are acceptable. As a consequence, Data Processing Module, Power Conversion Module, and Network Support Module should be given due attention if the conditions permit. Although elements 1 and 2 have higher uncertainty, they are located in the negligible region. Consequently, Graphics Processing Module can be ignored under limited conditions. If the conditions permit, in view of the higher uncertainty of elements 1 and 2 (Graphics Processing Module), by increasing the reliability of elements 1 and 2 and ensuring the reliability of the element's correlation with other elements, such as ensuring the reliability of the data transmission channel between elements 1 and 2 and other elements, and the reliability of the information transmission bus, etc. Based on these measures, the fault propagation from elements  
risk entropy, the critical risk elements are identified, which provides a theoretical basis for putting forward preventive measures, so as to ensure and improve system safety. Compared with the current methods and technologies, the method proposed in this paper mainly reflects the advantages of two aspects. On the one hand, the hierarchical model is modeled in a matrix manner, and the association relationship of each element in the complex system is quickly and accurately analyzed, which reduces the skill requirements of analysts. On the other hand, it provides a feasible and multi-faceted analysis method for the risk assessment of systems in view of fault propagation, which is the core judgment criterion for identifying critical risk factors and of great significance for ensuring system safety.  
Conceptualization, Z.J. and T.Z.; methodology, Z.J., S.W., and F.R.; formal analysis, Z.J. and F.R.; writing--original draft preparation, Z.J.; writing--review and editing, Z.J. and S.W.; project administration, T.Z.; funding acquisition, T.Z. and S.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Foundation of No.61400020404.  
The authors declare that there are no conflicts of interest.