## 4 Assessment Framework Based on Complex Networks and Fuzzy Petri Nets
### Dynamic Risk Quantification via FPN Simulation  
According to the FPR in Section 4.2, Table 1, and Figures 2 and 3, the FPN shown in Figure 5 can be obtained. :  
Building upon the FPN modeling principles detailed in Section 2 and referencing the works of [PERSON] et al. [33] and [PERSON] et al. [32], this framework additionally requires initialization of the following parameters: Transition thresholds \(\upmu\), Transition certainty factors \(\text{cf}_{\text{\tiny{$\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \}}}}}}}}}}}}}}}}\), Arc weights \(\text{w}_{\text{\tiny{$\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ }}}}}}}}}}}}}}}}}\) and Initial marking vector \(\text{M}_{0}\). Following the parameter configuration by [PERSON] et al. [33], we set \(\upmu\) = 0.05 and \(\text{cf}\) = 1 uniformly. The arc weight \(\text{w}\) has two calculation methods: (1) weights for node/edge-to-path vulnerability inference, determined by nodal/edge importance; (2) weights in AND-logic for protection chain places, determined by asset value or stakeholder priorities in the PPS. For method (1), Section 3.1 presents our importance calculation: let \(\text{x}_{\text{i}}\) denote the importance value of the corresponding edge/node of place \(\text{p}_{\text{i}}\) (\(\text{x}_{\text{i}}\) may take the value of either \(ND_{i}\) or \(\text{S}_{\text{i}}\)), then the arc weight to transition \(\text{t}_{\text{j}}\) is  
\[w_{ij}=\frac{x_{i}}{\sum_{j=1}^{M}x_{j}} \tag{7}\]  
Figure 5: FPN corresponding to the FPR shown in Section 4.2. \begin{table}
\begin{tabular}{l l} \hline
**Level of Inference** & **Contents** \\ \hline \multirow{4}{*}{Node/Edge \(\rightarrow\) Path} & IF d\({}_{1}\) and d\({}_{7}\) and d\({}_{3}\) and d\({}_{9}\) and d\({}_{4}\) and d\({}_{10}\) and d\({}_{5}\) Then d\({}_{12}\) \\  & IF d\({}_{1}\) and d\({}_{7}\) and d\({}_{3}\) and d\({}_{9}\) and d\({}_{4}\) and d\({}_{11}\) and d\({}_{6}\) Then d\({}_{13}\) \\  & IF d\({}_{2}\) and d\({}_{8}\) and d\({}_{3}\) and d\({}_{9}\) and d\({}_{4}\) and d\({}_{10}\) and d\({}_{5}\) Then d\({}_{14}\) \\  & IF d\({}_{2}\) and d\({}_{8}\) and d\({}_{3}\) and d\({}_{9}\) and d\({}_{4}\) and d\({}_{11}\) and d\({}_{6}\) Then d\({}_{15}\) \\ \multirow{4}{*}{Path \(\rightarrow\) Chain} & IF d\({}_{12}\) or d\({}_{14}\) Then d\({}_{16}\) \\  & IF d\({}_{13}\) or d\({}_{15}\) Then d\({}_{17}\) \\ \multirow{2}{*}{Chain \(\rightarrow\) System} & IF d\({}_{16}\) and d\({}_{17}\) Then d\({}_{18}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Inference process of the PPS in Figure 1.  
The initial truth degree \(\text{m}_{i}\) for input places is calculated as  
\[m_{i}=1-\frac{y_{i}}{10} \tag{8}\]  
where \(\text{y}_{i}\in[0,\,10]\) represents the protection score of subzone i (Section 4.1). Non-input places initialize to \(\text{m}_{i}=0\). The classical computation method for output place truth degrees typically uses matrix operations to calculate the output vector \(\text{M}_{n}\) at system stability from the initial vector \(\text{M}_{0}\)[32, 33]. However, as the number of places in FPN increases, the computational complexity of matrix operations grows significantly, making this approach impractical for FPN modeling and inference in complex PPS. To address this limitation, we adopt the method proposed by [PERSON] et al. [34], which computes vulnerability propagation values layer-by-layer at the transition level, enabling the efficient derivation of output place truth degrees. The FPN output place truth degree \(\text{m}_{n}\in[0,\,1]\) indicates higher PPS vulnerability as the values increase.