## 4 Simulation Experiment and Analysis  
### Experimental Explanation  
Through experimental simulation of the cascading failure model in the air traffic control network, the Network X library in Python [28] was used in the simulation network to explore and analyze network properties. Previously, most complex networks, including air traffic control networks, were considered random networks, with node degree distributions following the Poisson distribution. However, the research by [PERSON] and [PERSON] [29] showed that the Internet, computer wiring diagrams, and other networks are different from random networks, with node degree distributions following a power-law distribution and the existence of hub nodes. [PERSON] et al. [30] studied the topological characteristics of the global aviation network and found that it is a scale-free heterogeneous network. They also improved the node contraction method and ranked the importance of nodes. The simulation experiment network topology uses the BA model to generate the control coordination network \(G_{c}\) with power-law degree distribution and scale-free characteristics, and the air route facility network \(G_{a}\) is modeled based on the route map. The control coordination network \(G_{c}\) is modeled as a scale-free network with node degree distribution following a power-law distribution:  
\[P(\alpha)\sim\alpha^{-\lambda}, \tag{5}\]  
Here, \(P(\alpha)\) is the probability that a node has \(\alpha\) links, and \(\lambda\) is the power-law exponent. The topology of the control coordination network in the air traffic control system is similar to that of real communication networks, such as the Internet. During the generation process of the control coordination network, nodes are iteratively added, and each new node is connected to existing nodes. We denote the number of air route facility network nodes \(N_{a}\) that a control node \(V_{cm}\) can manage as the control capacity of the control node, denoted as \(n\). In the construction of  
Figure 6: Scenario of cascading failure model in air traffic control network. the control coordination network \(G_{c}\) in this article, it is assumed that the control capacity of each control node \(V_{cm}\) is consistent, and each control node in the control coordination network supports \(n\) nodes in the route facility network. After the construction of the control coordination network \(G_{c}\) and the air route facility network \(G_{a}\), the dependency relationship between the two networks is established. The method of constructing the dependency relationship is as follows: each node \(V_{a}\) in the air route facility network has \(k\) control dependency relationships from control nodes, and each control node can manage \(n\) nodes in the route facility network. Each node \(V_{c}\) in the control coordination network requires one service dependency relationship from a route node \(V_{al}\) in the air route facility network, and all service dependency relationships are randomly assigned. The nodes \(V_{a}\) in the air route facility network all have control dependency edges from the control nodes \(V_{cm}\), and the control coordination network nodes \(V_{c}\) all have service dependency edges from the route nodes \(V_{al}\). When the simulation network was established, the meanings of the node numbers used were as shown in Table 2.  
In order to effectively reflect the control capability of the air traffic control network, the control cost \(\theta\) is defined as the cost required for the control nodes \(V_{cm}\) to manage and control the air route facility network nodes \(V_{a}\):  
\[\theta=k\times N_{a}=n\times N_{cm}, \tag{6}\]  
Here, \(\theta\) represents the control cost of the air traffic control network, which represents the actual meaning of the number of control dependency edges in the air traffic control network. \(k\) represents the control cost required for a single air route facility network node \(V_{a}\), and \(n\) represents the control capability of a single control node \(V_{cm}\).