## 5 Example and Validation
### Illustrative Example
\begin{table}
\begin{tabular}{c l} \hline \hline
**Level of Inference** & **Contents** \\ \hline \multirow{7}{*}{\(\text{Node}/\text{Edge}\rightarrow\text{Path}\)} & IF d\({}_{1}\) and d\({}_{10}\) and d\({}_{4}\) and d\({}_{16}\) and d\({}_{6}\) and d\({}_{20}\) and d\({}_{8}\) then d\({}_{22}\) \\  & IF d\({}_{1}\) and d\({}_{10}\) and d\({}_{4}\) and d\({}_{17}\) and d\({}_{7}\) and d\({}_{21}\) and d\({}_{9}\) then d\({}_{23}  
Based on Equations (1), (2), and (7), the weights of the libraries in the FPN are computed, and the results are presented in Table 9:  
Figure 7: FPN model diagram for security system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(w_{ij}\) & **Value** & \(w_{ij}\) & **Value** & \(w_{ij}\) & **Value** \\ \hline \(w_{1,1}\) & 0.111 & \(w_{2,5}\) & 0.111 & \(w_{3,9}\) & 0.111 \\ \(w_{4,1}\) & 0.171 & \(w_{4,5}\) & 0.171 & \(w_{4,9}\) & 0.171 \\ \(w_{6,1}\) & 0.152 & \(w_{6,5}\) & 0.152 & \(w_{6,9}\) & 0.152 \\ \(w_{8,1}\) & 0.124 & \(w_{8,5}\) & 0.124 & \(w_{8,9}\) & 0.124 \\ \(w_{10,1}\) & 0.141 & \(w_{12,5}\) & 0.141 & \(w_{14,9}\) & 0.141 \\ \(w_{16,1}\) & 0.162 & \(w_{16,5}\) & 0.162 & \(w_{16,9}\) & 0.162 \\ \(w_{20,1}\) & 0.138 & \(w_{20,5}\) & 0.138 & \(w_{20,9}\) & 0.138 \\ \(w_{1,2}\) & 0.111 & \(w_{2,6}\) & 0.111 & \(w_{3,10}\) & 0.111 \\ \(w_{4,2}\) & 0.171 & \(w_{4,6}\) & 0.171 & \(w_{4,10}\) & 0.171 \\ \(w_{7,2}\) & 0.152 & \(w_{7,6}\) & 0.152 & \(w_{7,10}\) & 0.152 \\ \(w_{9,2}\) & 0.124 & \(w_{9,6}\) & 0.124 & \(w_{9,10}\) & 0.124 \\ \(w_{10,2}\) & 0.141 & \(w_{12,6}\) & 0.141 & \(w_{14,10}\) & 0.141 \\ \(w_{17,2}\) & 0.162 & \(w_{17,6}\) & 0.162 & \(w_{17,10}\) & 0.162 \\ \(w_{21,2}\) & 0.138 & \(w_{21,6}\) & 0.138 & \(w_{21,10}\) & 0.138 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Weighting of the route protection failure input place. Using Appendix A Table A1 and Equation (8), the value of the input place is derived, as presented in Table 10:  
After the FPN calculation, the truth degree obtained for Places P\({}_{22}\)-P\({}_{36}\) is shown in Table 11: Truth degree of places P\({}_{22}\)-P\({}_{36}\). Comparative analysis of Tables 4-7 and 11 yields three critical inferences:  
1. The system exhibits moderate vulnerability (m\({}_{36}\) = 0.406), necessitating scheduled risk mitigation measures. 2. Among all protection chains, c\({}_{2}\) (associated with the data center) demonstrates the highest vulnerability, surpassing c\({}_{1}\) (critical room). Data centers should be better protected. 3. Intrusion path r\({}_{10}\) presents the most critical risk among 12 identified paths, warranting prioritized reinforcement of its constituent components. 4. Node n\({}_{4}\) exhibits the highest vulnerability score of 0.575, approaching high vulnerability, which indicates insufficient protective capabilities in the parking lot. The PPS network diagram reveals that half of all intrusion paths include node n\({}_{4}\), demonstrating its high importance within the overall PPS. Given the parking lot's high \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Place** & **Truth Degree** & **Place** & **Truth Degree** & **Place** & **Truth Degree** \\ \hline \(p_{1}\) & 0.40 & \(p_{8}\) & 0.15 & \(p_{15}\) & 0.55 \\ \(p_{2}\) & 0.35 & \(p_{9}\) & 0.20 & \(p_{16}\) & 0.45 \\ \(p_{3}\) & 0.45 & \(p_{10}\) & 0.575 & \(p_{17}\) & 0.40 \\ \(p_{4}\) & 0.575 & \(p_{11}\) & 0.55 & \(p_{18}\) & 0.375 \\ \(p_{5}\) & 0.50 & \(p_{12}\) & 0.55 & \(p_{19}\) & 0.425 \\ \(p_{6}\) & 0.30 & \(p_{13}\) & 0.60 & \(p_{20}\) & 0.325 \\ \(p_{7}\) & 0.35 & \(p_{14}\) & 0.575 & \(p_{21}\) & 0.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Truth degree of input places.