# An Attention-Based Deep Convolution Network for Mining Airport Delay Propagation Causality
## 2 Problem Formulation  
The causal mining of airport delay propagation is to find the delay influence relationship between airports and to depict the extent to which delays at one airport lead to delays at another airport. Specifically, if delays at one airport come first, delays at the other airport come after. At the same time, the first delayed airport has an impact on the later delayed airport and the later delayed airport changes with the first delayed airport. It is considered that there is a delay causal relationship between the two airports, and the delay at the former airport is the cause of the delay at the latter airport. Recently, researchers have given a definition of causality between pairs of airports, that is, if the current and previous delay information of one airport helps explain the delay of another airport at a certain time in the future, then there is a causal relationship between them [15; 31; 32]. As shown in Figure 1, delay causality between airports can be summarized into four forms: no causation, direct causation, indirect causation, and both direct and indirect causation. Figure 1 shows four forms of airport delay propagation causality. In Figure 1a, a delay at airport A will not affect airport B. In Figure 1b, there is a direct causal relationship between airport A and airport B without intermediary airports, and the delay propagation path is \(\mathrm{A}\rightarrow\mathrm{B}\). In Figure 1c, there is an indirect causal relationship between airport A and airport B as an intermediate airport C, and the delay propagation path is \(\mathrm{A}\rightarrow\mathrm{C}\rightarrow\mathrm{B}\). It is important to note that there may be multiple indirect causal paths between two airports. In Figure 1d, there are both direct causal paths \(\mathrm{A}\rightarrow\mathrm{B}\) and indirect causal paths \(\mathrm{A}\rightarrow\mathrm{C}\rightarrow\mathrm{B}\) between airport A and airport B. Since the airport network is a network connected by air routes, there may be multiple indirect causal paths between any two airports. Direct causation involves a relationship between two airports, while indirect causation involves three or more airports. From the perspective of air traffic managers, direct causality to guide traffic control is more intuitive and easier to implement than indirect causality. Therefore, this section mainly focuses on the direct causality in the propagation of airport delays, that is, the direct impact of delays at one airport on delays at other airports, which does not consider the indirect effects of delay propagation on subsequent airports. In order to better represent the causal relationship of an airport network, a weighted directed graph is used to represent the causal relationship network of airport delay propagation. Suppose there are \(N\) airports in the airport network. Denote \(G=(\mathrm{V},\mathrm{E},\mathrm{D})\) as the airport delay propagation causality network; \(\mathrm{V}=\left\{v_{i}\right\}_{i=1;N}\) represents the airport set, where \(v_{i}\) represents the feature vector of the airport \(I\); \(\mathrm{E}=\left\{e_{ij}\right\}_{i,j=1;N}\) represents the set of edges in the airport network. If there is a direct causal relationship between the airport \(i\to j\), that is, the airport \(i\) has an edge pointing to \(j\), then \(e_{ij}=1\); otherwise, \(e_{ij}\)= 0. \(\mathrm{D}=\left\{d_{ij}\right\}_{i,j=1;N}\) represents the weight set of edges, where \(d_{ij}\) represents the weight of edge \(e_{ij}\), which means the degree of influence of the airport \(i\) on the airport \(j\). Figure 2 is a schematic diagram of the direct causality network of airport delay propagation, in which the arrows point to the direction of delay propagation, and the delay and the degree of influence are given above the arrow. For any two airports, the delay propagation between them can be divided into three cases: there is no delay propagation, the delay propagation is one-way, and the delay propagation is two-way.