## 5 Example and Validation
### Illustrative Example  
To validate the efficacy of the proposed vulnerability assessment method, this section conducts a case study on a representative security system. Following the analytical framework established in Section 3.1, the system is spatially and functionally divided into four defense layers with the following configurations:  
1. **Perimeter**: includes one personnel access point, one vehicle passage, and perimeter fencing. 2. **Surveillance Zone**: comprises a parking lot and an open plaza. 3. **Protection Zone**: houses critical assets across two buildings (A and B). 4. **Restricted Zone**: contains an archive room in Building A and a data center in Building B. This multilayered defense architecture exemplifies typical security system configurations, effectively demonstrating the method's applicability to complex systems. Subsequent subsections perform vulnerability assessment using the integrated complex network and FPN framework. Based on the system's topological structure, Figure 6 illustrates its complex network representation. \begin{table}
\begin{tabular}{l l l} \hline \hline \(\text{m}_{n}\) & **Vulnerability** & **Recommended Action** \\ \hline \([0,\,0.2)\) & Low & Maintain operations \\ \([0.2,\,0.4)\) & Moderate-Low & Implement risk mitigation \\ \([0.4,\,0.6)\) & Medium & Schedule urgent mitigation \\ \([0.6,\,0.8)\) & High & Immediate remediation required \\ \([0.8,\,1]\) & Critical & Cease operations; emergency response \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Vulnerability ratings. The FPR propositions obtained accordingly and their correspondence with the FPN Places are shown in Table 7:  
The inferential relationship between propositions is as shown in Table 8:  
This leads to the FPN model shown in Figure 7: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**p/d** & **Proposition** & **p/d** & **Proposition** \\ \hline \(p_{1}/d_{1}\) & Node \(n_{1}\) failure & \(p_{19}/d_{19}\) & Edge \(e_{9}\) failure \\ \(p_{2}/d_{2}\) & Node \(n_{2}\) failure & \(p_{20}/d_{20}\) & Edge \(e_{10}\) failure \\ \(p_{3}/d_{3}\) & Node \(n_{3}\) failure & \(p_{21}/d_{21}\) & Edge \(e_{11}\) failure \\ \(p_{4}/d_{4}\) & Node \(n_{4}\) failure & \(p_{22}/d_{22}\) & Path \(r_{1}\) failure \\ \(p_{5}/d_{5}\) & Node \(n_{5}\) failure & \(p_{23}/d_{23}\) & Path \(r_{2}\) failure \\ \(p_{6}/d_{6}\) & Node \(n_{6}\) failure & \(p_{24}/d_{24}\) & Path \(r_{3}\) failure \\ \(p_{7}/d_{7}\) & Node \(n_{7}\) failure & \(p_{25}/d_{25}\) & Path \(r_{4}\) failure \\ \(p_{8}/d_{8}\) & Node \(n_{8}\) failure & \(p_{26}/d_{26}\) & Path \(r_{5}\) failure \\ \(p_{9}/d_{9}\) & Node \(n_{9}\) failure & \(p_{27}/d_{27}\) & Path \(r_{6}\) failure \\ \(p_{10}/d_{10}\) & Edge \(e_{1}\) failure & \(p_{28}/d_{28}\) & Path \(r_{7}\) failure \\ \(p_{11}/d_{11}\) & Edge \(e_{2}\) failure & \(p_{29}/d_{29}\) & Path \(r_{8}\) failure \\ \(p_{12}/d_{12}\) & Edge \(e_{3}\) failure & \(p_{30}/d_{30}\) & Path \(r_{9}\) failure \\ \(p_{13}/d_{13}\) & Edge \(e_{4}\) failure & \(p_{31}/d_{31}\) & Path \(r_{10}\) failure \\ \(p_{14}/d_{14}\) & Edge \(e_{5}\) failure & \(p_{32}/d_{32}\) & Path \(r_{11}\) failure \\ \(p_{15}/d_{15}\) & Edge \(e_{6}\) failure & \(p_{33}/d_{33}\) & Path \(r_{12}\) failure \\ \(p_{16}/d_{16}\) & Edge \(e_{7}\) failure & \(p_{34}/d_{34}\) & Chain \(c_{1}\) failure \\ \(p_{17}/d_{17}\) & Edge \(e_{8}\) failure & \(p_{35}/d_{35}\) & Chain \(c_{2}\) failure \\ \(p_{18}/d_{18}\) & Edge \(e_{9}\) failure & \(p_{36}/d_{36}\) & System failure \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Places and vulnerability propositions.