## 3 The Fault Propagation and Instances of Cascading Failure Models  
Under normal circumstances, the air traffic control network maintains a regular operational state, and the control coordination network \(G_{c}\) and the air route facility network \(G_{a}\) support each other. According to the cascading failure model of the air traffic control network, the nodes in the network need to meet the following conditions to maintain normal operation: The node function is intact; the nodes in the air route facility network \(V_{a}\) have control dependency edges; the nodes in the control coordination network \(V_{c}\) have service dependency edges; and they are connected to the largest connected component. The process of cascading failure in the air traffic control network is shown in Figure 4. The process of cascading failure is as follows:  
1. The attacker attacks a node in the control coordination network \(G_{c}\) or the air route facility network \(G_{a}\) and causes it to fail;
2. Nodes that have dependency and internal link relationships with the failed node will also fail, and their normal operation conditions will be checked;
3. The failure of the node is then fed back to the corresponding control coordination network \(G_{c}\) and air route facility network \(G_{a}\). This process is iterated repeatedly until there are no new node failures, and the network reaches its final stable state, and the process of cascading failure ends. To further illustrate the control dependency relationships, service dependency relationships, and failure rules in the cascading failure model of the air traffic control network, an example is presented in this section to analyze the evolution of cascading failures in the network. In this example, as shown in Figure 5, there are two layers of networks, the control coordination network \(G_{c}\) and the air route facility network \(G_{a}\). The control coordination network \(G_{c}\) at the top contains two control nodes, \(C1\) and \(C2\), and four coordination nodes, \(M1\), \(M2\), \(M3\), and \(M4\). The air route facility network at the bottom contains three route nodes \(T1\), \(T2\), and \(T3\), and five facility nodes \(I1\), \(I2\), \(I3\), \(I4\), and \(I5\). The control dependency relationship between the control nodes and the nodes in the air route facility network is as follows:  
\[T1\to C1\;\;T2\to C1\;\;I1\to C1\;\;I4\to C1\]  
\[T3\to C2\;\;I2\to C2\;\;I3\to C2\;\;I5\to C2,\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Code Name** & **Meaning** \\ \hline \(G_{c}\) & Control coordination network. \\ \(V_{c}\) & Nodes in the control coordination network. \\ \(V_{cm}\) & Control nodes. \\ \(V_{ch}\) & Coordination nodes. \\ \(E_{c}\) & Edges in the control coordination network. \\ \(G_{a}\) & Air route facilities network. \\ \(V_{a}\) & Nodes in the air route facilities network. \\ \(V_{al}\) & Air route nodes \\ \(V_{ai}\) & Facility nodes. \\ \(E_{a}\) & Edges in the route facilities network. \\ \(k\) & The number of control nodes that the nodes in the route facilities network depend on. \\ \(n\) & Control capacity of the control nodes. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Codes and their meanings in the model. The service dependency relationship between the route nodes \(V_{al}\) and the control coordination network nodes \(V_{c}\) is as follows:  
\[M1\to T1\;\;M4\to T1\]  
\[C1\to T2\;\;M2\to T2\]  
\[C2\to T3\;\;M3\to T3\]  
In the air traffic control network, attackers generally use methods such as ADS-B injection to launch network attacks, due to the dispersed and location-differentiated characteristics of route nodes. Control nodes usually have certain protective measures, and it is difficult to directly attack nodes in the control coordination network. This paper mainly  
Figure 4: The cascading failure process diagram of the air traffic control network. Figure 5: Example of cascading failure model in air traffic control network.