# Situation Assessment of Air Traffic Based on Complex Network Theory and Ensemble Learning
## 2 Preliminaries
### Situation Assessment Indexes of Air Traffic Network  
In order to represent the status of the air traffic network, we chose 7 typical evaluation indexes in complex network analysis. The calculations of these indexes were based on \(w_{i}\). Therefore, they can comprehensively evaluate the topology and operation characteristics of air traffic network. The \"effective distance\" was introduced to replace the actual topological distance in the weighted network [26]. \"Effective distance\" can be calculated by the following formula:  
\[D_{ij}=1-\ln w_{ij} \tag{8}\]  
where \(w_{ij}\) is the weight of the connection between node \(i\) and node \(j\). Evaluation indexes of the complex network are calculated by \"effective distance\". They can reflect the actual operation characteristics of the air traffic network. The larger the network edge weight, the smaller the effective distance. The network average effective distance \(ND\) is:  
\[ND=\frac{1}{|E|}\sum_{i,j\in V}D_{ij} \tag{9}\]  
On this basis, the optimized network evaluation indexes were given as follows:  
1. Node Strength (\(NS\)) The weighted node degree is the node strength. \[\overline{s_{i}}=\frac{1}{N}\sum_{j=1}^{N}a_{ij}w_{ij}\] (10)  
The larger the \(NS\) value, the closer the link between the waypoint and the surrounding nodes. It can reflect the operation complexity of the current air traffic. Figure 1: The air traffic network in East China \(G_{2}(V_{2},E_{2},\omega_{2})\). (b) Node (edge) Betweenness (\(NB/EB\))  
It reflects the centrality of nodes and edges in the whole network. The number of interfaces \(B(v_{k})\) of node \(k\) refers to the proportion of the shortest effective distance passing through node \(k\) in the total effective distance:  
\[B(v_{k})=\frac{D^{\prime}{}_{ij}(v_{k})}{D_{ij}{}^{\prime}} \tag{11}\]  
where \(D_{ij}{}^{\prime}(v_{k})\) is the shortest effective distance between node \(i\) and node \(j\) through node \(k\), and \(D_{ij}{}^{\prime}\) is the shortest effective distance between node \(i\) and node \(j\). The edge betweenness is:  
\[B(e_{l})=\frac{D_{ij}{}^{\prime\prime}(e_{l})}{D_{ij}{}^{\prime}} \tag{12}\]  
where \(D_{ij}{}^{\prime\prime}(e_{l})\) is the shortest effective distance through edge \(l\). (c) Network Clustering Coefficient (\(NCC\))  
In the weighted network, the average clustering coefficient describes the aggregation characteristics of the network [19; 20]. The weighted clustering coefficient of a single node can be expressed as follows:  
\[c_{w}(i)=\frac{1}{s_{i}(k_{i}-1)}\sum\limits_{jk}\frac{w_{ij}+w_{ik}}{2}\cdot a _{ij}a_{jk}a_{ki} \tag{13}\]  
where \(s_{i}\) is the point strength, \(k_{i}\) is the node degree value, \(v_{j}\) and \(v_{k}\) represent two adjacent nodes of \(v_{i}\). \(a_{ij}\) indicates the connection status of node pairs. When \(v_{i}\) and \(v_{j}\) are interconnected, \(a_{ij}=1\), otherwise \(a_{ij}\)= 0. The larger the clustering coefficient of a single node, the more likely the node is at the core of the regional community. The average weighted clustering coefficient of the network is:  
\[\overline{C}=\frac{1}{N}\sum\limits_{i=1}^{N}c_{w}(i) \tag{14}\]  
It can reflect the clustering level of the network. (d) Network Efficiency (\(NE\))  
In the weighted network, the improved \(NE\) is the average of the reciprocal sum of the effective distances between all nodes:  
\[NE=\frac{1}{N(N-1)}\sum\limits_{i\
eq j}1/D_{ij} \tag{15}\]  
where \(D_{ij}\) is the effective distance between \(v_{i}\) and \(v_{j}\). It can reflect the difficulty of network information transmission. The smaller the \(NE\), the smoother the information transmission, and the stronger the network robustness. (e) Network Density (\(ND\))  
In the weighted network, the improved \(ND\) is:  
\[ND=\frac{\sum\limits_{i}^{n}\sum\limits_{j}^{n}a_{ij}wij}{2n} \tag{16}\]  
where \(n\) is the total number of network nodes. The greater the \(ND\), the higher the heterogeneity of the whole network.