# Situation Assessment of Air Traffic Based on Complex Network Theory and Ensemble Learning
## 2 Preliminaries  
### Construction of Air Traffic Network Model  
In the undirected graph \(G(V,E)\), \(V\) represents all nodes in graph \(G\), and \(E\) represents edges between all nodes in graph \(G\)[25]. In the air traffic operation network \(G_{1}(V_{1},E_{1},\omega_{1})\), \(V_{1}\) represents all nodes in graph \(G_{1}\), \(E_{1}\) represents edges between all nodes in graph \(G_{1}\), and \(\omega_{1}\) represents the edge weight. The nodes are airports and waypoints in \(G_{1}(V_{1},E_{1},\omega_{1})\). Edges are routes. The edge weight is calculated by the operation data of air traffic. Four typical operation indexes of air traffic were finally selected to characterize the operation situation of airports, waypoints, and routes. They were route saturation (\(RS\)), flight delay rate (\(FDR\)), meteorological conditions (\(MC\)), and military flights (\(MF\)). \(RS\) represents the ratio of the daily flow to the maximum capacity of routes. \(FDR\) represents the ratio of the number of delayed flights on the daily airports, waypoints, and routes to the number of normal flights. \(MC\) represents weather conditions for daily airports, waypoints, and routes. \(MF\) represents the ratio of the daily time of military activities affecting the operation of the airports, waypoints, and routes to the total time of normal operation. We sorted the \(MC\) into five categories including sunny, cloudy, rainy, foggy, and thunderstorm. The \(MC\) was presented in text format. In order to calculate these, we encoded the weather information in a 1-5 order. On this basis, we used the AHP method to construct the edge weight of air traffic network. Its process was simple, efficient, and applicable. According to the degree of influence on air traffic, the four indexes were ranked. \(RS\) is the core index reflecting the transport efficiency of the routes. It can indicate the operation situation of the routes. \(FDR\) is a supplement of \(AS\) in terms of delay. It can reflect the operational stability of the airports, waypoints, and routes. \(MC\) and \(MF\) are important factors that affect the normal flights. Compared with other indexes, \(MC\) and \(MF\) can indirectly reflect the operational situation of the airports, waypoints and routes. According to the statistics of the civil aviation administration, in recent years, military activities have a greater impact than meteorological conditions. Therefore, the order of importance of these four indexes is: \(RS>FDR>MF>MC\). The importance of the four indexes was compared by the scale method. The comparison results are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \(CV\) & \(RS\) & \(FDR\) & \(MF\) & \(MC\) \\ \hline \(RS\) & 1 & 3 & 5 & 7 \\ \(FDR\) & 1/3 & 1 & 3 & 5 \\ \(MF\) & 1/5 & 1/3 & 1 & 3 \\ MC & 1/7 & 1/5 & 1/3 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of importance. According to Table 1, the judgment matrix \(\mathbf{A}\) is:  
\[\mathbf{A}= \begin{bmatrix}1&3&5&7\\ 1/3&1&3&5\\ 1/5&1/3&1&3\\ 1/7&1/5&1/3&1\end{bmatrix} \]  
Calculating the eigenvector \(\omega\) corresponding to the maximum eigenvalue \(\lambda_{\text{max}}\), we perform normalization processing to obtain a weight vector:  
\[W_{i}=\sqrt[n]{\prod_{j=1}^{n}a_{ij}}\Bigg{/}\sum_{i=1}^{n}\sqrt[n]{\prod_{j= 1}^{n}a_{ij}} \tag{1}\]  
where \(a_{ij}\) is the element of \(\mathbf{A}\), and \(W=\left[W_{1},W_{2},\cdots,W_{n}\right]^{T}\) is the eigenvector corresponding to the maximum eigenvalue of \(\mathbf{A}\). According to Formula (1), the weight vector is calculated as follows:  
\[ \begin{array}{c}W=\left[\begin{array}{cccc}W_{1}&W_{2}&W_{3}&W_{4}\end{array} \right]\\ =\left[\begin{array}{cccc}0.5650&0.2622&0.1175&0.0553\end{array}\right] \end{array}