Airspace sectors in idle or busy time segments change air posture as the number, speed and heading of aircraft change. In order to accurately assess the airspace traffic and conflict situation, and to divide the sector to meet the requirements of busy/idle time operation, six topological indicators reflecting the local and global information of the airspace: total node degree, average node degree, average point strength, average weighted aggregation coefficient, network density, and network efficiency are selected as the influencing factors to divide the idle/busy time period.
3.1. Selection of Assessment Indicators
In order to comprehensively assess and classify the air traffic operation posture, six complex network characteristic indicators that can reflect the characteristics of the airspace conflict network from multiple perspectives are selected to describe the flight posture.
- 1.
Total Node Degree
Degree is a key attribute of a node that indicates the number of links between that node and other nodes. In a conflict network, the degree value reflects the number of conflicts between one aircraft and other aircraft, and most intuitively reflects the local conflict situation, denoted by
:
is the degree value of node i, N is the total number of nodes in the network, denotes the connectivity of two nodes, if connected and otherwise.
The total node degree reflects the total number of conflicts in the network;
is denoted by:
- 2.
Average Node Degree
On the basis of the degree value, for networks, the average degree is an important attribute of the network, indicating the average number of links between each node and other nodes throughout the network. In a conflict network, the average degree value can reflect the average number of conflicts each aircraft forms with its surrounding aircraft, denoted by
:
- 3.
Average Point Intensity
Point strength is the expression of the degree value after weighting; the index can not only reflect the number of neighboring nodes, but also reflect the degree of influence on the node around the node. In the conflict network, the average point strength can reflect the urgency of the conflict, and indirectly represents the average pressure of control deployment, which is denoted by
:
is the connection relationship between two nodes, if connected, otherwise; is the edge right between nodes , .
- 4.
Average Weighted Aggregation Factor
The weighted aggregation coefficient is a weighted expression of the ordinary aggregation coefficient, and a single value indicates the degree of aggregation at the location of a node and the degree of proximity between nodes, while the average weighted aggregation coefficient can reflect the degree of aggregation of the whole network. In a conflict network, this value can describe the degree of conflict aggregation of the network as a whole, which is denoted by
:
is the weighted aggregation factor, calculated as follows:
is the degree of node ; is the point strength of node ; is the number of connected triples; , are the neighbor nodes of node .
- 5.
Network Density
In the flight conflict network, the network density can be used to describe the denseness of the interconnecting edges between the nodes in the network, and the value can reflect the degree of saturation to the flight posture, denoted by
:
L is the actual number of connected edges in the network.
- 6.
Network Efficiency
The network efficiency visualizes the connectivity of the entire network; the efficiency between two nodes is expressed as the reciprocal of the distance to each other, while the efficiency of the entire network is the average of the efficiencies between every two pairs of nodes. In the conflict network, this value can easily represent the spreading breadth of the aircraft conflict, defined as:
is the shortest path between two nodes , . The higher the network efficiency NE, the closer the connection between the nodes, and the network is relatively more complex.
3.2. Integrated Network Indicator Classification
The six assessment indicators do not affect the flight attitude independently, but interact with each other and jointly affect the good or bad flight attitude at different times. The multi-factor interaction matrix (MFIM) was first applied to geological system engineering [
18]. It not only considers the influence of each parameter on the system, but also reflects the influence and interaction between multiple factors. It is an effective method for the analysis of complex system problems.
The MFIM method mainly includes three processes: (1) encoding the multi-factor interaction matrix; (2) Calculate the subjective and objective influence degree of each index; (3) Allocate the weight of each index. This method can fully explore the relationship between indicators and analyze the internal factors of complex systems. However, the expert scoring method is usually used in matrix coding. This method is too subjective and lacks scientificity to some extent. Therefore, we wish to improve the coding method. The first step is to place the influencing factors on the main diagonal of the matrix (the order can be interchanged).
The maximal information coefficient is an effective method for determining the correlation between two variables [
19]. Most correlation analysis finds it difficult to explore the nonlinear relationship between variables [
20]. However, MIC can not only find linear and nonlinear correlations in the data, but also explore the potential non-functional correlations between them. If there is a connection between the two variables, the network partition of the scatter diagram composed of the two points can always have a suitable division method to reflect its relevance. The specific description is as follows:
Suppose that there exists a data set D, which contains two two-dimensional data variables
x and
y; then, the mutual information of
x and
y can be expressed as:
In the formula, is the joint probability distribution of x and y; increases with the increase in the correlation between x and y.
Since the calculation of the joint probability distribution is difficult, the scatter plot of the
x-
y data is meshed, and each continuous
x (or
y) is divided into the corresponding column
x group (or row
y group), thereby obtaining a new
X,
Y grouping. The mutual information calculation based on this is as follows:
Further, the correlation between the two variables
MIC is defined as:
In the formula, the value of B is the 0.6 power of the sample size N, and the value of MIC is in the range of [0,1]. A higher MIC value indicates that the correlation between variables is stronger.
In addition, the position of the triangle on the main diagonal of the matrix (the blue part of the graph) is filled with the evaluation value A, and the Delphi 0–1 scoring method [
21] is used to invite multiple experts to conduct independent scoring and evaluation. Describe the strength relationship between the indicators; the closer the value is to 1, the more important it is, and the less important it is.
The evaluation value on the non-main diagonal line of each row indicates the influence of the main diagonal factor on other factors, that is, the subjective effect. The code on each column of non-main diagonal indicates that the main diagonal is affected by other factors, that is, the objective effect. For example, represents not only the subjective influence of factor on factor in the horizontal direction, but also the objective influence of factor on factor in the vertical direction. Therefore, the multi-factor interaction matrix is asymmetric.
According to the construction principle of interaction matrix, the degree of subjective and objective effects of influencing factor
i can be calculated according to the following formula:
In the formula, n is the number of influencing factors; and are the degree of subjective effect and objective effect of factor i in MFIM, respectively. and are the sum of the subjective and objective effect weights of all influencing factors; is the weight value of each index combined with subjective and objective influence.
The MATLAB R2020b software is used to simulate the conflict network generated by the airspace in
Figure 3, which contains 75 nodes. After visualization, the depth of the node color is set according to the degree value. The larger the degree value is, the deeper the color is, indicating that the conflict is more serious. The thickness of the edge represents the urgency of the conflict. The thicker the edge is, the more urgent the conflict is. The computational complexity metrics are shown in
Table 1.
The multi-factor mutual matrix of the six indicators calculated by this method is shown in
Figure 4. The blue part is the expert scoring weight, and the green part is the maximum information coefficient weight.
According to the MFIM method, the final weights and the order of importance of the six indicators were
(2.86) >
(2.71) >
(2.63) >
(2.30) >
(2.16) >
(1.85). Each evaluation indicator was normalized to obtain a constant number of weights. The weighting of the combined six indicators constitutes the comprehensive network metrics (CNM) for evaluating the flight posture, and the CNM can be expressed as:
This index can comprehensively describe the flight conflict situation and flight posture in the airspace, using MATLAB software to simulate the airspace to generate 100 conflict networks that reflect the air operation situation. The conflict network setting rules are as follows: randomly generate 10–100 aircraft nodes, use the above aircraft protection area model and three-dimensional velocity obstacle method to judge the conflict relationship between aircraft, calculate the indicators in different conflict networks as sample sets, and calculate the comprehensive network indicator values of each conflict network. At the same time, three experts in related fields are invited to evaluate the flight situation and divide it into two different periods: busy/idle. The indicators and expert scoring are shown in
Table 2.
The comprehensive network index is corresponding to the time-period division. The specific numerical division of the evaluation index is bounded by the value of 4. The comprehensive network index is greater than 4 and is defined as the busy period, and less than 4 is defined as the idle period.