## 2 Cascading Failure Model for Air Traffic Control Networks
Route nodes \(V_{al}\) are geographical nodes on the aircraft route, such as position reporting points, mandatory reporting points, airports, navigation points, important points, transition points, holding points, control transfer points, control permission boundary points, etc. They are mainly key geographical nodes, established on the aircraft route for navigation, reporting aircraft positions, and changing air traffic service settings. Facility nodes \(V_{ai}\) are relevant physical service facilities in the physical network, such as aircraft, ground-air communication facilities, satellite communication facilities, monitoring facilities, navigation equipment, meteorological service facilities, terminal information facilities, etc. They are mainly physical facilities, such as sensors, detectors, and transducers, with the function of collecting surrounding information. The edges \(E_{a}\) of the air route facility network represent the transmission of sensing information data between nodes in the air route facility network \(G_{a}\). After modeling the information network and physical network, we establish edge relationships between the control coordination network \(G_{c}\) and the air route facility network \(G_{a}\). Considering the interdependence within the air traffic control network, two special dependency edge relationships are defined as control dependency and service dependency. In the air traffic control network, control nodes \(V_{cm}\) have the ability to control all nodes \(V_{a}\) in the air route facility network. Control dependency refers to the dependence of each node \(V_{a}\) in the air route facility network on \(k\) control nodes \(V_{cm}\) in the control coordination network \(G_{c}\). Each control node \(V_{cm}\) has a control capability of \(n\), which means that the operation of route points and related physical service facilities in the air route facility network are managed and controlled by various control positions. At the same time, control coordination nodes \(V_{c}\) in the control coordination network require service data support provided by each route node \(V_{al}\) in the air route facility network \(G_{a}\). Due to the difference in address location, all service data is uniformly uploaded to control coordination network nodes \(V_{c}\) by route nodes \(V_{al}\). Service dependency refers to the dependence of control coordination network nodes \(V_{c}\) on route nodes \(V_{al}\) in the air route facility network \(G_{a}\), indicating that control and coordination positions in the control coordination network are supported by service data from each route point. These two dependency relationships are called dependency edges. A dependency edge \(U\to V\) represents a one-way dependency edge from node \(U\) to node \(V\). It means that the dependency edge \(U\) only works when \(V\) is running, but \(V\) does not depend on \(U\). This dependency relationship is directional and asymmetric. If node \(U\) depends on node \(V\), it does not mean that node \(V\) depends on node \(U\). Only when there is a bidirectional dependency edge between nodes \(U\) and \(V\) is it indicated that \(U\) and \(V\) depend on each other to work properly. This is shown in Figure 3. In this paper, both control dependency and service dependency are mandatory dependency relationships. The control dependency of the route facility network nodes depends on the control nodes, which reflects the air control in the air traffic management system. The service dependency of the control coordination network nodes depends on the route nodes, which reflects the control and sharing of information resources in the air traffic management system. Figure 3: Dependency edge. When establishing the cascading failure model, the meanings of the node codes are shown in Figure 1:0