## 3 Theoretical Foundations
### Complex Network Theory  
Complex network theory serves as a fundamental theoretical framework for studying the topological structure and dynamic behavior of large-scale, nonlinear interactive systems, with its core method employing graph theory models to characterize inter-component relationships [22; 23]. Formally, a complex network can be defined as a weighted directed graph \(G=(N,E)\)[24; 25], where  
1. Node set \(N\) represents physical or functional components. 2. Edge set \(E\) denotes inter-component relationships with weights quantifying interaction intensity or propagation likelihood. In public transportation planning, researchers have adopted the Space-L complex network modeling method to analyze bus routes and stations. This method constructs topological models of transit networks by treating bus stations as nodes and adjacency relationships between stations as edges [26; 27]. Similarly, in PPSs, protection zones and their connecting pathways can be abstracted as nodes and edges in complex networks, making the Space-L method applicable for protection system modeling. The core principle states that an edge exists between two nodes if direct or indirect connecting paths exist between them, while no edge is established for nodes without physical connections in real environments [28]. This modeling approach effectively characterizes PPS topology, providing a foundation for subsequent vulnerability analysis. Figure 1a shows a schematic diagram of a physical protection system (PPS), where red lines indicate inter-zone connecting paths. As illustrated in Figure 1a, nodes n\({}_{1}\) and n\({}_{2}\) represent vehicle and pedestrian access points on the perimeter, providing access to n\({}_{3}\) (an outdoor area within the PPS) and, subsequently, to n\({}_{4}\) (internal building passageways), ultimately leading to zones n\({}_{5}\) and n\({}_{6}\) (rooms housing critical assets). Figure 1b demonstrates its complex network representation, where directional edges encode reachability relationships between zones. Figure 1: Example of physical protection system modeling under Space L rule. (**a**) shows a schematic diagram of a physical protection system (PPS), where red lines indicate inter-zone connecting paths. As illustrated in (**a**), nodes n\({}_{1}\) and n\({}_{2}\) represent vehicle and pedestrian access points on the perimeter, providing access to n\({}_{3}\) (an outdoor area within the PPS) and, subsequently, to n\({}_{4}\) (internal building passageways), ultimately leading to zones n\({}_{5}\) and n\({}_{6}\) (rooms housing critical assets). (**b**) demonstrates its complex network representation, where directional edges encode reachability relationships between zones. In complex networks, heterogeneous node/edge significance necessitates quantified characterization. This study employs the following statistically validated metrics to evaluate topological criticality [29]: (1) Degree Centrality (\(K\)) The degree \(K_{i}\) of node \(i\) refers to the number of other nodes that are directly connected to it. It reflects the level of connectivity of the node within the network. A higher degree indicates that the node has more connections with other nodes, implying a greater involvement or centrality in the network structure. (2) Average Path Length (\(L\)) Average path length is defined as the mean value of the shortest paths between all node pairs in the network. In our study, the complex network contains designated source and target nodes. We specifically define this metric as the average length of the shortest directed paths from all source nodes to the target node. (3) Betweenness Centrality (\(B\)) Node betweenness represents the frequency with which a node appears in all shortest paths. For our framework, we calculate node betweenness as the proportion of source-to-target shortest paths that pass through a given node. Higher betweenness values indicate greater nodal influence within the network. (4) Node Importance (\(ND\)) Node importance measures the significance of nodes within the network.