The Construction of an Aircraft Control Multilayer Network and Its Robustness Analysis

When aircraft are fying along the routes, their states present diversity and complexity. To clarify the control or monitoring relationship between air sectors and aircraft in the airspace and to promote reliability and efciency of air trafc control (ATC), in this paper, an aircraft control multilayer network (ACMN) model is built by considering the air sector network (ASCN) and the aircraft state network (ASTN). Te characteristics of ACMN are studied based on complex network theory. Simultaneously, according to the multilayer characteristics of ACMN and the relative entropy theory, a robustness analysis method is proposed. Results show that the proposed method is applicable for evaluating the structural stability and robustness of ACMN and is efective in improving the efciency of ATC. It can also provide a new idea for the robustness research of multilayer networks.


Introduction
Air transport is an important part of the trafc system. In recent years, more and more people have chosen air travel, and the aviation industry has ushered in great development. As a result, air trafc busyness and congestion are rising, and the safety, reliability, and smoothness of air trafc are facing challenges. When an aircraft is fying along the route, its distribution is complex and diverse, and the state between aircraft is bound to have relevance. Each aircraft can be regarded as a point, and the correlation between aircraft can be expressed by a line. Tus, an aircraft state network model is formed, and the key points in the model can be efectively evaluated so as to analyze the correlation complexity of aircraft. Terefore, the state of the aircraft can be accurately handled, which can not only improve the smoothness and safety of fight but also improve the accuracy and efciency of air trafc control (ATC).
In addition, aircraft fy in air sectors, and each sector is monitored by air trafc controllers. Considering the air sector network and aircraft state network, the aircraft control multilayer network (ACMN) model is conducive to clarifying the relationship between air sectors and aircraft, thereby promoting the safety and reliability of ATC. Te ACMN model is essentially a complex network, so its characteristic analysis can be based on complex network theory. Simultaneously, according to the multilayer characteristics of ACMN and the relative entropy theory, a network robustness analysis method is proposed in this paper. Te results can provide a reference for improving the efciency of ATC.
Tis paper is organized as follows: the literature review is in Section 2, and Section 3 analyzes the air-state relationship among aircraft and establishes the ACMN model. Section 4 introduces some basic centrality theories and establishes a robustness analysis model. Te results are analyzed in Section 5. Finally, Section 6 concludes this paper and future work.

Literature Review
Te aircraft control multilayer network (ACMN) is essentially a complex network model, and its node evaluation analysis can be implemented based on complex network robustness of the air transport network. Pien et al. [31] conducted topological and operational analyses of the European air trafc network (ATN) in order to derive a more relevant and appropriate indicator of robustness. To analyze the temporal pattern of the Chinese aviation network (CAN), a time series of topological statistics through sliding the temporal CAN with an hourly time window is obtained in [32]. Te robustness of the air route network (ARN) is analyzed by the topology potential and relative entropy methods in [33]. In addition, Ren and Li [34] compare the characteristics of air trafc networks between China and the United States. Over the past two years, the robustness of aviation networks has also received attention [35,36].
Based on the research above, previous papers rarely analyze air trafc networks from the perspective of ATC. In this paper, the aircraft control multilayer network will be established, and then its characteristics and robustness will be analyzed to provide a reference for the robustness analysis of the aviation network. Furthermore, to promote the safety and smoothness of ATC.

Aircraft State Network (ASTN)
3.1. Air Route Network. Airspace is the specifc space in which aircraft operate in the air. Te trajectory of aircraft in the airspace is guided by the air route; that is, the air route is the carrier of aircraft. Te utilization efciency of airspace is deeply afected by the structure of air routes, and the crowding efect of airspace. Route structure and aircraft distribution are of practical signifcance for ensuring aviation safety and improving airspace utilization efciency. In this section, the structural characteristics of the route and the correlation of aircraft along the route will be analyzed, and then the aircraft state network (ASTN) model will be established.
Te airline network represents the transportation relationship between the initial airport and the destination, which is a logical relationship between OD pairs. It refects the fight direction, origin, destination, and stop location of the aircraft. However, unlike the airline, the air route represents the physical trajectory of the aircraft. In general, the aircraft does not fy along the optimized and smooth trajectory between airports in the airspace but follows the preset path network in the airspace. Te diagram of the air route network (ARN) is shown in Figure 1.
As shown in Figure 1, route sections and waypoints (such as navigation points, route intersections, turning points, and reporting points) are connected to each other to form an ARN model. In general, the ARN has a hub-andspoke structure. Furthermore, the structure of ARN shows that the most basic ones are linear, cross, convergent, and divergent. Te specifc structure of ARN is shown in Figure 2.
Air trafc fow is distributed on the ARN, and the distribution of trafc is unbalanced. As air trafc continues to grow, there is an increasing likelihood of fight conficts, delays, and localized congestion in the air. Te structural characteristics of ARN have an important infuence on the smooth operation of the aircraft in airspace, and they also afect the distribution state of aircraft. Te distribution of aircraft along diferent air routes and the correlation between them play an important role in the efciency and orderliness of aircraft in the airspace.

Establishment of ASTN.
Te air state of the aircraft refers to the distribution and operation trend of the aircraft when it is running in the airspace. Tere is a situational correlation between two aircraft when there is a direct impact or connection between them, such as a fight confict, delay, or congestion. Obviously, the state relevance of aircraft is directly afected by fight direction, fight attitude, and route structure. Te distribution of aircraft along diferent air routes is shown in Figure 3.
As shown in Figure 3(a), three aircraft A 1 , A 2 , and A 3 fy along the air route. A 3 is the front plane, and its fight condition directly afects the rear A 2 , while A 2 afects A 1 . Terefore, there is a state correlation between A 2 and A 1 or A 3 (represented by red dotted lines), and this correlation has a transmission efect. In Figure 3 Te aircraft can be regarded as nodes, and an edge is added if there is a state correlation between the nodes, thus constructing the ASTN model. Te ASTN along diferent air routes is shown in Figure 4.
. . , n represents the number of nodes; that is, the number of aircraft, E � (v i , v j )|i ≠ j, v i , v j ∈ V is the number of edges, which represents the state correlation between aircraft. Te state correlation between aircraft can be expressed by A � (a ij ) n×n , and the elements in the matrix are a ij � 1, State correlation between aircraft i and j, For example, in Figure 4(b), its adjacency matrix is expressed as Te ASTN refects the smooth operation of aircraft in airspace, in which the key nodes (aircraft) have an important impact on the safety, efciency, and stability of fight operation. Accurate and efective identifcation of key nodes in the model is of great signifcance for improving airspace utilization and reducing delay. Moreover, the robustness of ASTN can refect the orderliness of air operations, which is important for air trafc control (ATC).

Air Sector Network.
Airspace is the specifc environment for aircraft operation, and it is also the area where aircraft receive air trafc control services. In order to improve the reliability and efciency of ATC and ensure the safety and order of aircraft operations, the airspace is divided into several areas, each of which is equipped with at least one air trafc controller; this area is called air sector. Each sector is a basic ATC unit. When the aircraft passes through the sector, the controller and the aircraft will have a control connection, and at the same time, there will be communication between adjacent sectors.
Each sector is reduced to a node. If there is communication between two sectors, an edge is added, which constitutes an air sector network (ASCN) model. Te ASCN model is shown in Figure 5.

Aircraft Control Multilayer Network.
Air route sections and waypoints are widely distributed in the sector. At the same time, the airways can guide the fight of the aircraft. When aircraft are fying, the controller will always pay attention to them and communicate with them. Within a certain period of time, all aircraft in the sector are controlled by the sector controller. Besides, a sector will communicate with multiple aircraft, so there is a "one-to-many" relationship between the sector and the aircraft. Terefore, the aircraft control multilayer network (ACMN) has the characteristics of a dual-level and multicorresponding relationship.
Assuming that the ACMN is expressed as G, the ASTN is G1, and the ASCN is G2, then, Based on the abovementioned analysis, the adjacency matrix A of G includes the following parts.
For G1(or G2), the element of adjacency matrix A (G1) is For the adjacency matrix A (G1G2) between two layers, its elements are represented as Terefore, the adjacency matrix of ACMN is Te structure of ACMN is shown in Figure 6. It can be seen from Figure 7 that aircraft A1, A2, and A3 are in sector 1, so they are connected to sector 1, forming a multilayer network relationship. Similarly, other aircraft are also connected to their corresponding sectors. Finally, the multilayer network model of an aircraft is formed.

Basic Parameters.
Te ACMN is a complex network, so the basic parameters of complex network theory can be used. Node centrality refects the relative importance of each node in the model. In this paper, the node centrality of the ACMN indicates that there are direct connections between the aircraft and their surroundings. Te key aircraft directly afects the operation state of others, which also play an important role in the safety and smoothness of airspace.
Te main parameters that represent node centrality in complex network theory are degree, betweenness, and closeness centrality.
Te degree centrality C D (v i ) of node v i can be expressed as where d i is the degree of node v i , (n − 1) is the maximum degree. In ASTN, degree centrality represents the tightness of one aircraft with its neighbors, refecting the operational situation (smoothness, safety, etc.) of the aircraft. Similarly, in ASCN, the degree of centrality represents the connectivity of a sector to its neighboring sectors, refecting the importance of the sector in the airspace.
Te degree centrality C D of network G can be expressed as where where B i represents the betweenness of node v i . In ASTN, the betweenness centrality refects those changes in the operational situation of one aircraft that have a greater impact on the operation of other ones. Meanwhile, the betweenness centrality of one sector indicates the importance of the sector to the global structure and stability of the Journal of Advanced Transportation airspace. Of course, closeness and centrality have similar meanings.
Te betweenness centrality C B of network G is where C B (v max ) is the maximum betweenness of nodes in G.
Te closeness centrality C C (v i ) of node v i is  where l ij represents the distance from node v i to v j . Te closeness centrality C C of G is where C C (v max ) is the maximum closeness of nodes in G.

Characteristic Parameters.
Te cumulative degree is used to represent degree distribution in complex networks, and its expression is where P d denotes the probability distribution of nodes whose degree is not less than d.
In addition, the degree-degree correlation (r) of the network is also an important statistical feature. Generally, it can be expressed by the Pearson correlation coefcient and the relationship between the degree and the average degree of the nearest neighbor.
Te degree-degree correlation is where d i and d j are the degrees of the two nodes of edge e ij , E is the set of edges, and M is the number of edges. Te absolute value |r| ∈ [0, 1], r > 0 represent the positive correlation of the network, r < 0 represents the negative correlation, and r � 0 represents the noncorrelation. Te nearest-neighbor average degree is where a ij is the element of the adjacency matrix. Furthermore, the average value of d nn,i is expressed as where N is the number of nodes, P(d) is the degree distribution. d nn (d) is an increasing function of d, indicating that the network is assortative. Otherwise, the network is disassortative.

Evaluation Index.
Te average path length L(G) of G is expressed as where l ij represents the distance between any two nodes. Te average reciprocal of the distance between nodes is network efciency, expressed by E(G), and then, Network connectivity can refect the integrity of a certain function, and the impact of nodes on network connectivity can refect the relative importance of nodes. In this section, the maximum connected component and connectivity coefcient are used to measure the connectivity of the network.
Te ratio C R (v i ) is expressed as where n is the number of nodes in the network, and n v i is the number of nodes in the maximum connected component after removing node v i .

Multilayer Network Robustness Model.
When a node in a multilayer network is deleted, there are two changes in the single-layer network and cascading failures between networks. Terefore, when analyzing its robustness, the comprehensiveness of the multilayer network should be considered, and the change in the robustness index should be considered as a whole. Let G be a multilayer network with m layers and the network efciency is [E(G1), E(G2), . . . , E(Gm)]. When the network G is attacked, the remaining network can be expressed as G ′ . After a random attack, the efciency of each layer of G ′ is [E(G r ′ 1), E(G r ′ 2), . . . , E(G r ′ m)], and after a deliberate attack, the efciency of each layer of G ′ is [E(G d ′ 1), E(G d ′ 2), . . . , E(G d ′ m)]. Te distribution of the abovementioned network efciency can be expressed as Te abovementioned data can be regarded as a distribution of random variables. Te robustness of a multilayer network can be represented by the total relative change in the efciency of diferent layers. Terefore, the relative entropy theory can be used to measure the network efciency change.
Let P(x) and Q(x) be two random distributions, and their relative entropy D can be expressed as For the network efciency distribution of multilayer networks, the relative entropy can be expressed as

Journal of Advanced Transportation
Ten, the relative entropy ΔE is normalized as follows: Furthermore, for multilayer networks, the robustness index based on network efciency can be expressed as Similarly, in the multilayer network G, the maximum connected component size of each layer is For the distribution of the maximum connected component size of a multilayer network, its relative entropy ΔR can be expressed as Similarly, the relative entropy ΔR is normalized as follows: Terefore, for multilayer networks, the robustness index based on the maximum connected component size can be expressed as Finally, for the ACMN, its robustness can be represented by the combination of BE and BR.

Data Analysis. Te Controlled Airspace of Beijing (BCA) is one of the busiest areas of civil aviation in China.
Based on the statistics of the Air Trafc Administration, this section takes the BCA as an example for analysis. Te spatial distribution of air routes and sectors in the BCA is shown in Figure 8.
In Figure 8, the spatial distribution of air routes and sectors in the BCA is shown. Te red solid line represents the boundary of the controlled airspace, and the red dots represent key waypoints. Tese key waypoints include navigation points, position reporting points, route intersections, etc., and routes are between the waypoints. In addition, the blue solid line represents the boundary of the air sector, and the numbering shows that the number of sectors is 16.
Te distribution of 72 aircraft in BCA within a time window is shown in Figure 7. According to the situation correlation analysis in Section 1, the ASTN model can be established as shown in Figure 9.
According to the distribution of aircraft, the control relationship between air sectors and aircraft is analyzed, and an aircraft control multilayer network (ACMN) model is constructed. Te results are shown in Figure 10.

Characteristic Analysis.
Te ACMN is a typical complex network, so its basic characteristics can be analyzed by complex network theory. Te cumulative degree distribution function is used to describe the degree distribution characteristics, and the results are shown in Figure 11. Figure 11 shows that the degree distribution of ACMN obeys an exponential distribution and its ftting degree R 2 > 0.9. Terefore, the degree of the ACMN decreases rapidly; that is, the degree of a few nodes is large, and the degree of most nodes is relatively small. Furthermore, the degree-degree correlation of the ACMN is analyzed, and the degree correlation coefcient (r) is calculated. Te results are shown in Figure 12. Figure 12 shows the changes of the nearest-neighbor average degree d nn (d) with the degree d. Te trend shows that the nearest-neighbor average degree has a positive correlation with the degree; that is, d nn (d) is an increasing function of d. In addition, the degree-degree correlation coefcient r > 0 of the ACMN shows that nodes with a large degree tend to connect large-degree ones. Terefore, the ACMN has assortative structure characteristics.

Journal of Advanced Transportation
Ten, the basic characteristics of each subnet in the ACMN model are given, and the results are shown in Table 1.
It can be seen from Table 1 that the number of nodes in ASTN is 72 and the number of edges is 197, so its density is 0.07, which is a relatively low density. Te average degree is 3, indicating that on average, the state connection among each of the three aircraft is the closest. From the clustering coefcient and the average shortest path, the ASTN has no obvious small-world characteristics. Te density of the ASCN is 0.41, indicating that the network is relatively dense, and its average degree is 3, indicating that each aircraft fies through an average of 3 sectors. At the same time, the ASCN exhibits certain small-world characteristics. Similarly, the ACMN is a low-density network, and every 5 nodes are most closely connected. In addition, the ACMN model does not have obvious small-world characteristics.

Robustness Analysis Based on Attack Strategy.
In a complex network, the centrality of a node expresses the important infuence of a node's attributes in the network. In the ASTN, the node is the aircraft, and its centrality has an important relationship with the congestion, delay, and safety of the aircraft in the airspace. Namely, the relationship between the aircraft is that during the fight, the state of each aircraft will be afected by the others (for example, when one aircraft accelerates or climbs, adjacent or even surrounding ones may have to monitor or make avoidance operations). In the ASCN, the centrality of nodes refects the control relationship between aircraft and the sector. Terefore, different nodes have diferent efects on the robustness of the ACMN, and attacking diferent nodes can refect the robustness of the model.
Next, the attack strategy is implemented on the ACMN. First, the nodes are sorted from high to low according to the results of node centrality, and then the nodes are attacked in turn to obtain the evaluation index value. Te robustness of ACMN is analyzed based on complex network theory, and the results are shown in Figure 13. Figure 13 shows the changes in network efciency and the maximum connected component under an attack strategy. Te trend of the curve is consistent with the common complex network, but it does not refect the infuence of the characteristics of the multilayer network on the robustness. Next, the robustness will be analyzed based on the characteristics of multilayer networks.

Robustness Analysis Based on Multilayer Networks.
For the ACMN, the multilayer network is intentionally attacked based on diferent centrality strategies, and then the robustness of the network model is analyzed. From the perspective of cascading failure in a multilayer network, when the air sector is closed by interference, the aircraft in it cannot accept the control instructions and cannot pass normally, so the ASTN is also afected, resulting in cascading failure. Te robustness is shown in Figure 14. A noncascading failure means that when the air sector is afected, the aircraft can communicate with each other through airborne navigation or other navigation methods so that the fight can be maintained. At this time, the ASTN can avoid the impact of the air sector. Its robustness is shown in Figure 15.
Te results in Figures 14 and 15 show that the integrity of the sector structure has an important impact on the stability of the ACMN, which is consistent with the rule that the normal fight of aircraft is regulated by the sector. Besides, the stability of the sector also has an impact on the reliability and efciency of the fight.
Next, based on the relative entropy theory in Section 4.4, the infuence of the coupling between ASTN and ASCN on the robustness of ACMN is analyzed, and the results are shown in Figure 16. Figures 16(a) and 16(b) show that with the failure of nodes, the relative entropy of the network efciency and the relative entropy of the maximum connected component both increase rapidly. Tat is, the more the network efciency and the maximum number of connected components are afected, the more obvious the relative entropy changes are, and the weaker the network can maintain stability. Te curves in Figures 16(c) and 16(d) show that betweenness and degree have the greatest impact on the robustness of the ACMN. Furthermore, the coupling properties of ASTN and ASCN are also refected, and the results demonstrate the practicability and efectiveness of the proposed method.

Conclusions
Tis paper analyzes the state of aircraft on the route in the airspace, as well as the relationship between each aircraft and the air sector, and builds the ACMN. Ten, considering the centrality methods of degree, betweenness, and closeness in complex network theory and introducing relative entropy theory, a robustness evaluation model based on a multilayer network is proposed. Finally, taking the state of the aircraft in BCA as an example, the ACMN model is established, and the centrality method is used to evaluate the centrality of the aircraft. Te results are verifed under deliberate attack and random attack strategies, with network efciency and maximum connected components as evaluation indicators. Te analysis shows that degree and betweenness have obvious efects on node centrality. Simultaneously, it shows that the proposed method is efective and reliable for the robustness analysis of multilayer networks. In fact, the stability of ASTN refects the orderliness and normality of the operation of the aircraft group in the airspace. In addition, the stability of ASCN represents the structural integrity of the airspace. Teir global stability is the guarantee of the safety and reliability of air transport, and of course, it is also the basis for ensuring the smoothness of ATC. So, an accurate analysis of the robustness and node centrality of the ACMN model is conducive to improving the safety, reliability, and orderliness of air transport and promoting ATC. Tis proposed model lacks consideration of the dynamic characteristics of aircraft; in the future, based on ATC, a real-time dynamic ACMN model (considering the dynamic characteristics of aircraft) can be established to optimize the distribution of trafc fow in the airspace, improve the efciency of the airspace, and reduce air delays.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that there are no conficts of interest.