## 2 Cascading Failure Model for Air Traffic Control Networks  
A Cyber-Physical System (CPS) [22] is a complex system that integrates sensing, computing, and control, with features such as real-time perception, control management, and information services. The air traffic management network is a complex cyber-physical system [23], formed by the coupling of information and physical networks, integrating facilities such as navigation, radar, communication, surveillance, and sensing, and providing services such as air traffic control, navigation information, air traffic safety, and airspace management, as shown in Figure 1. Currently, in the application of CPS in air traffic control, [PERSON] et al. [24] studied the application of critical tasks in CPS systems in the aviation network from three aspects: aircraft, aircraft operation control, and air navigation operation and maintenance services. [PERSON] et al. [25] established an aviation CPS network model, based on node attributes and mobile tasks. [PERSON] et al. [25] proposed an air traffic CPS model, based on control network and route network. In the air traffic control network, the information and physical network systems are composed of a large number of components, and, in most cases, the impact of most terminals on the system reliability is limited. Therefore, when modeling, only the component nodes that play an important role in the network are considered. Based on the air traffic CPS model [26], this paper considers the differences in node functions and proposes two types  
Figure 1: Air traffic control network. of dependencies. According to the civil aviation air traffic management regulations [27] and the interdependence between control units and physical route service facilities, the air traffic control network is modeled as a cascading failure model of interdependent complex networks, as shown in Figure 2.  
Within the information and physical networks, the internal links are undirected. When modeling, both networks are represented as undirected graphs. The information network is the control coordination network, which is modeled as a graph:  
\[G_{c}=(V_{c},E_{c}), \tag{1}\]  
\[V_{c}=V_{cm}\cup V_{ch}, \tag{2}\]  
Here, \(G_{c}\) is the control coordination network, and the node set \(V_{c}\) of the control coordination network is divided into control nodes \(V_{cm}\) and coordination nodes \(V_{ch}\). Control nodes \(V_{cm}\) are control positions in control units, such as airport control positions, ground control positions, approach control positions, procedural control positions, radar control positions, flow management positions, etc. They are mainly responsible for controlling activities of aircraft and reporting points within corresponding control sectors. Coordination nodes \(V_{ch}\) are coordination, operation, and support positions in control units, such as release permission release positions, notification coordination positions, director positions, military coordination positions, flight plan processing positions, search and rescue coordination positions, dynamic maintenance positions, etc. They are mainly responsible for reporting navigation dynamic information and plans to aircraft, relevant positions, and related units, and conducting necessary coordination. The edges \(E_{c}\) of the control coordination network represent the transmission of information instructions between connected positions in the control coordination network \(G_{c}\). The physical network is the air route facility network, which is modeled as a graph:  
\[G_{a}=(V_{a},E_{a}), \tag{3}\]  
\[V_{a}=V_{al}\cup V_{ai}, \tag{4}\]  
Figure 2: Cascading failure model for air traffic control networks. In this network, \(G_{a}\) is the air route facility network, and the node set \(V_{a}\) of the air route facility network is divided into route nodes \(V_{al}\) and facility nodes \(V_{ai}\).