## 5 Conclusions  
In this article, the air traffic control network was modeled using the theory of inter-dependent networks and CPS theory. A cascading failure model of the air traffic control network, composed of the control coordination network and the air route facility network, was constructed. The dependency relationships of control dependency and service dependency were proposed, and the propagation process of cascading failures under this model was analyzed. This model is closer to the actual operation of the air traffic control network and more clearly reflects the relationship between control and service support within the air traffic control network, as well as the process of cascading failures. The relationship between the resilience of the air traffic control network system and parameters such as the control capacity of control nodes, the number of control nodes, and the control cost was studied. The results showed that, under the condition of determining the topology of the air traffic control network, enhancing the control capacity of each control position can effectively improve the resilience of the network by reducing the rate of cascading failures; under the condition of determining the control capacity of control positions, increasing the number of control positions can also effectively improve the resilience of the network by reducing the rate of cascading failures; enhancing the control capacity and the number of control positions can both increase the control cost of the air traffic control network and also reduce the rate of cascading failures, which is conducive to improving the resilience of the network; compared to increasing the number of control positions, enhancing the control capacity of control positions is more conducive to improving the resilience of the air traffic control network. Degree attacks on route nodes have a greater impact on the cascading failure of the air traffic control network, compared to random attacks or degree attacks on facility nodes. The cascading failure model of air traffic control networks proposed in this article shows that once a cascading failure occurs in the air traffic control network, it may cause the entire air traffic control system to collapse, leading to serious impacts on flight safety and economy. Although increasing control costs can improve the safety of the air traffic  
Figure 12: Schematic diagram of the air traffic control network in North China. (**a**) Control coordination network in North China. (**b**) Air route facility network in North China. (There are 20 control nodes represented by circles in the control coordination network, and 69 air route nodes in the air route facility network, including 28 airports, represented by squares, and 41 other navigation nodes, represented by intersections of routes in the figure). control network, it also increases the operating costs of aviation authorities in operating the air traffic control network. However, compared with the safety and stability of the air traffic control system, these costs and resource inputs are necessary. Implementing the strategies suggested in the cascading failure model is feasible. Our work can help aviation authorities better understand the risks and safety issues in the air traffic control system, providing a reference for them to develop more scientific and effective safety management measures. Although aviation authorities may need some cost and resource inputs, these measures can effectively improve the safety and reliability of the air traffic control system, reduce flight delays and cancellations, enhance the robustness of the air traffic control network, ensure the time cost, economy, and safety of airlines and passengers, and bring safer and more convenient air traffic services to society. Although this article provides some ideas and methods for studying cascading failures in the air traffic control network, it has not yet considered the differences in control nodes under different sectors and the elasticity of the air traffic control network's own recovery.