## 4 Assessment Framework Based on Complex Networks and Fuzzy Petri Nets
### FPR-Driven Vulnerability Propagation Logic  
FPR employs fuzzy inference to model the causal logic of PPS failures. To enable fuzzy inference, we extend the complex network model of PPS beyond basic nodes (N) and edges (E) with the following definitions:  
1. **Intrusion Path (R):** Intrusion Path Set R comprises all possible paths (r\({}_{1}\)-r\({}_{n}\)) from potential entry points to critical asset locations, with each individual intrusion path r\({}_{m}\) representing a complete route from a specific entry point to a particular critical asset location. 2. **Protection Chain (C):** Protection Chain Set C consists of all protection chains (c\({}_{1}\)-c\({}_{n}\)). An individual protection chain c\({}_{m}\) is defined as the collection of all intrusion paths from potential entry points to a particular critical asset. A PPS containing multiple critical assets is considered to have multiple protection chains. Consider the network in Figure 1b: Restricted Zones n\({}_{5}\), n\({}_{6}\) store critical assets A, B, with n\({}_{1}\), n\({}_{2}\) as intrusion origins. This system contains four intrusion paths and two protection chains, as shown in Table 1.  
Based on FPR inference rules and system characteristics, we define four failure propositions:  
1. **Component Failure**: edge/node security compromise
2. **Path Failure**: security breach along an intrusion path R
3. **Chain Failure**: compromise of a protection chain c
4. **System Failure**: global security collapse  
In accordance with the established reasoning framework, Figure 1b's components are formally mapped as nodes (p\({}_{1}\)/d\({}_{1}\)-p\({}_{6}\)/d\({}_{6}\)), edges (p\({}_{7}\)/d\({}_{7}\)-p\({}_{11}\)/d\({}_{11}\)), paths (p\({}_{12}\)/d\({}_{12}\)-p\({}_{15}\)/d\({}_{15}\)),protection chains (\(\mathrm{p_{16}/d_{16}\text{-}p_{17}/d_{17}}\)), and system failure (\(\mathrm{p_{18}/d_{18}}\)). Table 2 exemplifies these propositions using the Figure 1b network. The inference process is outlined as follows: The vulnerability of edges and nodes, reflecting their likelihood of failure, enables the derivation of intrusion path vulnerability. This, in turn, allows the calculation of the vulnerability of the protection chain that encompasses these intrusion paths. Ultimately, the overall vulnerability of the PPS can be inferred from the vulnerabilities of these protection chains. **Node/Edge \(\rightarrow\) Path**: AND logic governs the inference, as all components must fail to compromise a path. **Path \(\rightarrow\) Chain**: OR logic applies, breaching one path suffices to compromise a chain. **Chain \(\rightarrow\) System**: Two mission-driven logics exist: (1) AND: System vulnerability scales with partial asset loss (e.g., redundant systems); (2) OR: Any chain failure triggers total system failure (e.g., critical infrastructure)  
For the Figure 1b case study, taking intrusion path \(\mathrm{r_{1}}\) as an example, if all security zones (\(\mathrm{n_{1}}\), \(\mathrm{n_{3}}\), \(\mathrm{n_{4}}\), \(\mathrm{n_{5}}\)) and connecting paths (\(\mathrm{e_{1}}\), \(\mathrm{e_{3}}\), \(\mathrm{e_{4}}\)) experience a protection failure, the entire path is considered compromised, which follows the \"AND\" rule in FPR. Table 1 presents the corresponding proposition numbers for these events. Thus, the complete causal logic for the failure of path \(\mathrm{r_{1}}\) is expressed in FPR as  
\(\mathrm{d_{1}}\) and \(\mathrm{d_{7}}\) and \(\mathrm{d_{3}}\) and \(\mathrm{d_{9}}\) and \(\mathrm{d_{4}}\) and \(\mathrm{d_{10}}\) and \(\mathrm{d_{5}}\) Then \(\mathrm{d_{12}}\)  
For protection chain \(\mathrm{c_{1}}\), intruders reaching critical room \(\mathrm{n_{5}}\) through either path \(\mathrm{r_{1}}\) or \(\mathrm{r_{2}}\) follow FPR's \"OR\" rule, expressed formally as  
\(\mathrm{d_{12}}\) or \(\mathrm{d_{13}}\) then \(\mathrm{d_{16}}\)  
Similarly, the complete inference process is as shown in Table 3: