A geographical and operational deep graph convolutional approach for flight delay prediction

Kiqun CAI, Yue LI, Yongwen ZHU, Qun FANG, Yng YANG,Weno DU,*

a School of Electronic and Information Engineering, Beihang University, Beijing 100191, China

b Key Laboratory of National Airspace Technology, Beijing 100085, China

c National Lab of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

d Research Institute for Frontier Science, Beihang University, Beijing 100191, China

KEYWORDS Flight delay prediction;Flight operation pattern;Geographical interactive information;Graph neural network;Spatial-temporal information

Abstract Flight delay prediction has attracted great interest in civil aviation community due to its significant role in airline planning,flight scheduling,airport operation,and passenger service.Flight delay is affected by numerous factors and irregularly propagates in air transportation networks owing to flight connectivity, which brings critical challenges to accurate flight delay prediction.In recent years, Graph Convolutional Networks (GCNs) have become popular in flight delay prediction due to the advantage in extracting complicated relationships.However,most of the existing GCN-based methods have failed to effectively capture the spatial–temporal information in flight delay prediction.In this paper, a Geographical and Operational Graph Convolutional Network(GOGCN) is proposed for multi-airport flight delay prediction.The GOGCN is a GCN-based spatial–temporal model that improves node feature representation ability with geographical and operational spatial–temporal interactions in a graph.Specifically, an operational aggregator is designed to extract global operational information based on the graph structure,while a geographical aggregator is developed to capture the similar nature among spatially close airports.Extensive experiments on a real-world dataset demonstrate that the proposed approach outperforms the state-of-the-art methods with a satisfying accuracy improvement.

1.Introduction

Air transportation plays an important role in business and tourism, serving nearly 4.5 billion passengers worldwide in 2019, yet with a delay of 13 min per flight.1These delays have inevitably negative impacts such as economic losses of airspace users, unpleasant passenger experiences, and other indirect financial losses.The annual cost of flight delays to the global economy was estimated to be$50 billion in 2019.2To alleviate such severe losses, flight delay prediction has received considerable attention from both industry and academia.

The existing methods for flight delay prediction can be divided into two categories: single-airport scenario and multi-airport scenario.Specifically, single-airport flight delay prediction typically extracts influential factors from an airport and utilizes machine-learning-based models(e.g.,linear regression,3random forests,4and support vector machines5) to predict future flight delays.However,the above works neglect the spatial interactions among airports.

Regarding the multi-airport scenario,researchers have proposed a series of methods(e.g.,network decomposition,6delay causality network,7and delay-tree framework8) to capture the propagation effects of flight delays in airport networks.However,the abovementioned works mainly analyze the qualitative influence of flight delays from historical data and cannot capture the quantitative spatial–temporal patterns of flight delays.

Recently,with the superior capability of extracting complicated relationships from graph-structured data, GCNs have been utilized to model the dynamic spatial–temporal interactions in networks.9–11Particularly, GCN-based methods have been widely applied in practical flight delay prediction with promising performance.Based on the direct air route connections in an airport network, researchers transform the multiairport flight delay prediction into a graph representation learning task and propose a series of deep learning architectures to predict future flight delays12–14.

Despite the advantages in understanding the spatial–temporal dependencies among airports in flight delay prediction,the existing GCN-based methods cannot effectively solve the following 2 problems.

(1) In regional airspace, several geographically neighboring airports are generally subject to the same air traffic management unit and similar local weather conditions,resulting in strong interrelationships among these airports.Indeed,most of the previous GCN-based methods ignore the spatial–temporal interactions between geographically adjacent airports without direct air route connections.

(2) Except for the regional geographic interrelationships among neighboring airports, the inherent air route structure is also critical for flight delay prediction, as it indicates the global flight operation pattern in an airport network.However, the existing GCN-based approaches cannot simultaneously capture the regional geographic interrelationships and the global spatial–temporal interactions in airport networks.

To address the abovementioned problems and improve the performance of flight delay prediction, this paper proposes a GOGCN for multi-airport flight delay prediction.Considering the regional geographic interrelationships among adjacent airports in bounded airspace,a geographical aggregator based on the Euclidean distance is proposed.Moreover, to simultaneously extract the regional geographic information and the global flight operation patterns hidden in airport networks, a comprehensive aggregator based on two types of aggregation technologies is developed in the proposed model.Overall, the proposed approach can be distinguished from prior works in the following 2 aspects.

(1) A novel geographical aggregator is proposed to capture the geographical interactions among adjacent airports and is shown to satisfy the permutation invariant property.

(2) A comprehensive aggregator is developed to simultaneously extract the regional geographic impact and global flight operation patterns in airport networks.The remainder of this paper is organized as follows:Section 2 summarizes the related works on flight delay prediction and graph neural networks.In Section 3,we formulate the multi-airport flight delay prediction problem.Section 4 describes the methodology, including deep learning architecture, operational aggregator, and geographical aggregator.In Section 5, the proposed method is compared with several benchmark approaches.Section 6 presents the case study of the Chinese airport network.Section 7 concludes the paper with brief remarks.

2.Related work

2.1.Flight delay prediction

Most of the existing flight delay prediction methods can be reviewed from 2 perspectives: single-airport scenario and multi-airport scenario.

In the single-airport scenario, the existing approaches for flight delay prediction mainly employ statistic-learning-based methods (e.g., linear regression, random forests, and support vector machines)to predict future flight delays based on extensive influential factors.Yu et al.15utilized a deep belief network combined with a set of micro influential factors to carry out a practical flight delay prediction at Beijing Capital International Airport.The complexity of flight delays restricts the abovementioned work from accurately extracting influential factors.Therefore,researchers learn the mapping relationship between complexity factors and flight delays from abundant operational data.Based on Automatic Dependent Surveillance-Broadcast (ADS-B) messages, Gui et al.16developed random-forest-based and Long Short-Term Memory(LSTM)-based architectures to predict individual flight delays.However, these approaches have a major drawback of ignoring complicated spatial interactions among airports which are critical for accurately predicting flight delays.

In the multi-airport scenario, researchers have tended to introduce the spatial dependencies of airport networks in flight delay prediction.Considering the topology characteristics and dynamical operation of an airport network, Wang et al.studied the flight delays with respect to connectivity.17,18Based on the interdependence of delay time series,Du et al.7employed a delay causality network to investigate flight delays in a largescale airport network.Considering multiple resources(e.g.,aircraft, crew, and passenger) spatial–temporal connections, Wu and Law8utilized a Bayesian network to investigate flight delay propagation in a delay-tree framework.To summarize the abovementioned works, the spatial–temporal correlations of flight delays have been abstracted by researchers based on the human experience.

Recently, several intelligent frameworks have been developed to learn knowledge from spatial-temporal operational data for flight delay prediction.Based on a GCN, Cai et al.12refined the flight delay prediction problem as a graph representation learning task and proposed a deep learning architecture to predict flight delays in an airport network.Considering the direct air route connections in an airport network,Bao et al.13developed a graph-to-sequence learning architecture to predict the hourly departure and arrival delay of the network.Based on the topological structure of an airport network, Ai et al.investigated the flight delays with respect to flight connectivity and proposed a deep learning method for multi-airport flight delay prediction14.

Despite the advances in understanding the spatial dependencies among airports in flight delay prediction, the mentioned works disregard the spatial interactions between geographically adjacent airports without direct air route connections.

2.2.Graph convolutional networks

GCNs,with the capability of learning latent representations on graph-structured data, have been successfully applied in various research areas.19–21GCN22and the Graph Attention Network (GAT)23are traditional deep convolutional learning frameworks that define the graph convolution operator as a weighted sum of the features of the connected neighbors.To improve the flexibility and scalability of the abovementioned aggregation schemes, Hamilton24and Jia et al.25developed more advanced aggregation techniques that inductively capture feature information and explicitly avoid redundant computation.However, these aggregation schemes neglect the interactions among neighbors.

Considering the interactions among connected neighbors,Zhu et al.26proposed a graph neural network architecture that defines the graph convolution operator as the weighted sum with pairwise interactions of connected neighbors.However,it fails to model the structural information of nodes in neighborhoods.Based on the geometric relationships defined in a latent space, Pei et al.27developed a bi-level aggregation technique to update the feature representations.Nevertheless,these methods have restrictions in modeling the complicated interactions in graphs and fail to extract the feature information of semantic neighbors.

Recently,to efficiently capture global and local spatial correlations in a graph,Lu28and Wang et al.29proposed superior deep convolutional learning models that learn feature representations from both spatial and semantic neighbors of nodes.Moreover, by constructing multilevel spatial–temporal subgraphs,Wu30and Li31et al.designed hierarchical frameworks to update node feature representations based on structural and functional neighbors.

Despite the advances in modeling spatial relationships hidden in a graph, the semantic neighbors in current works are assumed to be distant and independent, which hardly corresponds to reality.

3.Problem formulation

4.Methodology

4.1.Overview of proposed model

The main idea of the proposed model is to simultaneously capture regional geographic interactions and global flight operation patterns in airport networks.Fig.1 presents the framework of the proposed GOGCN, which mainly consists of 2 modules:Operational Aggregator(OA)and Geographical Aggregator (GA).

Specifically, the OA takes a time-evolving airport network and departure features as input and employs a localized firstorder approximation of spectral graph convolutions to learn the representation vector based on operational neighbors.Similarly, the GA takes departure features and the geographical continuous space underlying airport networks as input and extracts target node feature information by aggregating different nodes within a geographical neighborhood.

The main idea of the proposed approach is to fuse global operational pattern and regional geographical interactions in an airport network.OA employs a localized first-order approximation of spectral graph convolutions to learn the representation vector based on operational neighbors.GA extracts target node feature information by aggregating different nodes within a geographical neighborhood.ReLU represents REctified linear unit, which is a frequently used activation layer in deep neural networks.FC denotes a fully connected layer.YOAand YGArepresent the prediction results obtained by OA and GA, respectively.α and β are hyperparameters that weigh the strength of the OA and GA.

4.2.Operational Aggregator

An airport network (a directed graph) can be considered as a multi-relational graph with incoming and outgoing relations,where flights connect all airports.Moreover, all carriers execute similar daily flight plans, resulting in a periodic flight operation pattern that indicates the propagation mechanism of flight delays in airport networks.

Inspired by the Message-Passing Neural Network(MPNN),32–34which has been intensively utilized to capture complicated spatial–temporal dependency in real networks,we employ message-passing technology to capture the global flight operation pattern in an airport network.

In a layer of the MPNN,each node updates its feature representation by aggregating all information from the neighbors.The neighbors are often defined as the set of connected neighbors in a graph, and the objective of the MPNN is to learn a representation vector for each node.

Similarly,the OA obtains the representation vector of each node by recursively aggregating the features from operational neighbors in an airport network:

4.3.Geographical Aggregator

In air transportation, several geographically adjacent airports are generally regulated by the same air traffic management unit and shared similar local weather conditions.Therefore,due to strict regulations and geographic proximity, there is a strong interrelationship among these airports in regional airspace.

A limitation of the OA is that interactions between 2 geographically adjacent airports are not modeled.To learn the representation vector of each airport, besides the directly related operational neighbors, the geographical neighbors should also be considered even if an air route connection between them does not exist.

Fig.1 System architecture of the proposed method.

4.3.1.Operational similarity among airports

In air transportation, the operation patterns of an airport include increased demand for departure flights, reduced demand for departure flights,and constant demand for departure flights.It is obvious that the more similar the operation pattern between 2 airports is, the more competitive they are for departure resources.

To capture the regional geographic interactions among adjacent airports, the Euclidean distance is used to calculate the operation similarity between 2 airports.Accordingly, considering the time-evolving departure features at each airport,the Euclidean distance between airports u and v can be expressed as.where T equals the number of time points, and xvtand xutdenote the number of departure flights at time point t.

The smaller the Euclidean distance between 2 airports is,the more similar the operation patterns of the 2 airports are,namely the more competitive the two airports are.Accordingly, when extracting node feature information from neighbors, greater weights should be assigned to airports with a smaller Euclidean distance.

4.3.2.Geographical aggregation product

Based on the abovementioned discussion, the GA obtains the representation vector of each node by recursively aggregating the features from geographical neighbors:

4.3.3.Proof of permutation invariant

GA is a novel aggregator proposed in this paper for the flight delay prediction problem, which can learn the interactions between two geographically adjacent airports even if an air route connection between them does not exist.Meanwhile,permutation invariance is an important property for a new GNN aggregator.24Therefore,permutation invariant property should be proved for a novel aggregator.

The permutation invariant property of GA can be intuitively understood from Eq.(4): when the order of input vectors is changed, the permutation invariant property holds if all terms on the right of Eq.(4) do not change.26,35,36To provide strict proof, Eq.(4) can be rewritten in matrix form:

which reveals the permutation invariance.

4.4.GOGCN model

The main idea of the proposed approach is to fuse regional and global spatial–temporal features from geographical and operational neighbors.Specifically, a linear combination scheme is utilized to build the novel graph convolution operator:

where H (l) indicates the representation vector for all nodes at the lth layer, σ denotes a nonlinear activation function, and Afand Agrepresent the graph topology of the OA and GA,respectively.When α is set to be zero, the GOGCN only employs the GA to process the information based on geographical neighbors;when β is set to 0,no geographical neighbors are considered and the GOGCN degrades to the traditional GCN.

Since the OA and GA are permutation invariant,it is obvious that the new graph convolution operator defined by Eq.(7) is also permutation invariant.

In air transportation,the airport network exhibits a typical hub-and-spoke status leading to great differences in the absorption and diffusion of flight delays among different airports.The hub airport,representing the rare sample in the Chinese airport network, handles massive flight traffic based on superior infrastructures.However, these airports are characterized by highly fluctuating flight delays due to full-load operation.The spoke airport, occupying the vast majority of the Chinese airport network, handles limited flight traffic based on general infrastructures.And these airports are characterized by smooth flight delays and little difficulty in delay forecasting.Accordingly, to reduce the distinction among samples,29,37,38we develop a weighted loss function that assigns higher weights to rare samples (i.e., hub airports) in the training phase.All airports are divided into five levels according to handling capacity η (the cumulative passenger throughput in a calendar year), i.e., I= {0 ,1,2,3,4}.Specifically, i=0 denotes the 1st level of the airport that satisfies η ≥100 million, i=1 reveals the 2nd level of the airport that satisfies η ≥10 million,i=2 represents the 3rd level of the airport that satisfies 2 million ≤η< 10 million, i=3 indicates the 4th level of the airport that satisfies 1 million ≤η<2 million,and i=4 denotes the 5th level of the airport that satisfies η< 1 million.Y(i) is utilized to denote the airport with handling capacity level i.

where Y demonstrates real delay information, ^Y represents the prediction of the model, and λiis a hyperparameter that indicates the weight of the samples with handling capacity level i.

5.Experimental results

5.1.Dataset

To evaluate the proposed model,a real-world dataset provided by the Civil Aviation Administration of China is used with basic information shown in Table 1.The dataset consists of flight ID, planned/actual departure flight information,planned/actual landing flight information, flight delay information, and the corresponding time.The dataset contains approximately 1.8 million scheduled flights connecting 209 civil airports, which serve more than 95 % of the air traffic in China.

5.2.Experimental settings

To avoid the outlier issue, all data is normalized by the ZScore normalization method.39Additionally, considering the data fluctuation issue,all labels are processed through Kalman filtering.40Specifically, at each time point, Kalman filtering takes a sequence of average delay of 209 airports as input and employs the statistical properties of the original labels to estimate the new labels with minimal error.70%of the dataset is employed for training,15%of the dataset is utilized for testing and the remaining 15 % is used for validation.All experiments are tested on a Linux cluster (i.e., CPU: Intel (R) Xeon(R) Gold 6126 CPU @ 2.60 GHz, GPU: NVIDIA TITAN RTX).

The distance threshold ρ is set as 260 km.The length of the time interval is set as 1 h,and the time window is set as 15 min.The graph convolution kernel size is set to be 3, the initial learning rate is 1 × 10-4and the dropout rate is set to be 0.3.The hyperparameters α and β are set to be 0.5 and 0.7,respectively.The sample weights in the loss function are set as 0.5,0.25,0.2,0.1,and 0.05.Our models are trained by minimizing the weighted mean square error using the Adam optimizer for 50 epochs.

We vary α and β from 0.1 to 0.9 and test the impacts on prediction performance for the real-world dataset, as shown in Fig.2.When α grows from 0.1 to 0.5, Mean Absolute Error(MAE) decreases gradually on the dataset; with α increasing from 0.5 to 0.9, MAE increases slightly on the dataset.When β grows from 0.1 to 0.7, MAE declines gradually on the dataset; with β increasing from 0.7 to 0.9, MAE increases slightlyon the dataset.Therefore,we set α and β as 0.5 and 0.7,respectively, and GOGCN can achieve better and stable performance.

Table 1 Basic information of used real-world dataset.

We also vary the distance threshold ρ from 80 to 320 km and test the effect on prediction performance for the realworld dataset, as shown in Fig.2.When ρ grows from 80 to 260 km, MAE decreases drastically on the dataset.With ρ increasing from 260 to 320 km, MAE increases slightly on the dataset.Therefore, we set the distance threshold ρ as 260 km, and GOGCN shows better performance.

In addition,we vary the size of the time-window from 10 to 30 min and test the effect on prediction performance for the real-world dataset, as shown in Fig.2.When the size of the time-window grows from 10 to 15, MAE declines gradually on the dataset.With it increasing from 15 to 30, MAE increases slightly on the dataset.Therefore, we set the timewindow as 15 minutes, and GOGCN can achieve better and more stable performance.

The time interval,the graph convolution kernel size,the initial learning rate,and the dropout rate are set with reference to the benchmark model (i.e., MSTAGCN12).And we chose an Adam optimizer with the advantages of fast convergence and easy parameter tuning.

5.3.Evaluation metrics

The objective of this paper is to predict the average quantitative delay(i.e.,an amount of time that quantifies both the cancellations and departure flight delays) at each airport at a future time point.Specifically, the output of the proposed method is a sequence of 209 average quantitative delays at a future time point.Accordingly, MAE, Root Mean Squared Error(RMSE),and Mean Absolute Percentage Error(MAPE)are employed to evaluate the performance (i.e., accuracy) of the models.However, the MAPE hardly works when the real delay value yiis extremely small.Therefore, Symmetric Mean Absolute Percentage Error (SMAPE) is also employed for the evaluation.

where k is the number of testing samples; ^yjand yjdenote real delay information and predicted flight delay, respectively.

5.4.Baseline methods

To evaluate the performance of the proposed model, 8 related state-of-the-art approaches are selected as baselines:

Fig.2 Impacts of α, β, ρ and time-window on prediction performance.

(1) RF41: Random Forest.It can effectively handle highdimensional data and capture complicated non-linear relationships,which is one of the most popular methods in time-series prediction tasks.

(2) GCN22:Graph Convolutional Network.The GCN captures node feature information by a localized first-order approximation of spectral graph convolutions and is widely applied for graph-structured data.

(3) GAT23: Graph Attention Network.The GAT extracts node feature information by assigning different levels of importance to different nodes within a neighborhood and does not need to know the entire graph structure in advance.

(4) GraphSAGE24: A general inductive framework that leverages node feature information to efficiently generate node embeddings for previously unseen data.

(5) Geom-GCN27:Geometric graph convolutional network.Based on a continuous space underlying the graph, the GCN employs a geometric aggregation scheme to capture the long-range dependencies in disassortative graphs.

(6) BGCN26: Bilinear Graph Neural Network.The BGCN uses the weighted sum with pairwise interactions between two neighbor nodes to improve the GCN representation ability.

(7) MSTAGCN12:As the latest work of multi-airport flight delay prediction based on the GCN, the MSTAGCN develops spatial–temporal convolutional blocks to predict flight delays within incomplete, time-evolving,graph-structured inputs.

(8) GSNet29: Spatial-Temporal Geographical and Semantic Network.GSNet learns multiscale spatial–temporal dependencies from geographical and semantic neighbors for traffic accident risk prediction.

5.5.Experimental results

5.5.1.Model comparison

The prediction performance of the proposed approach and baseline methods on the testing datasets is listed in Table 2.We independently run each experiment 10 times and report the mean and standard deviation.It can be demonstrated that our model achieves the best performance in terms of all metrics.

Specifically, it is easy to observe that the GCN, GAT, and GraphSAGE do not perform well due to their limited ability to model the interactions among neighbors.Compared with the abovementioned aggregation scheme, the Geom-GCN,BGCN,MSTAGCN,and GSNet make further improvements.By mapping the original graph to a latent continuous space,the Geom-GCN develops a geometric aggregator to model the geometric interactions among neighbors.Based on the weighted sum with pairwise interactions of neighbor nodes,the BGCN explicitly encodes the local node interactions within a neighborhood.Through aggregating node information from connected airports, the MSTAGCN proposes two convolutional blocks to learn the time-evolving patterns of flight delays and the unknown occasional air route structure.Based on spatial–temporal convolutional modules, GSNet extracts node feature information by assigning different levels of importance to geographical and semantic neighbors.However,the abovementioned methods fail to jointly model regional geographic interactions and global operational information in real networks.

Overall, by simultaneously aggregating node feature information from operational neighbors and geographical neighbors, the proposed GOGCN can effectively capture both global and regional spatial–temporal relationships in airport networks, which contributes to improving the prediction accuracy.

5.5.2.Ablation study

To illustrate the effectiveness of each component in the proposed GOGCN, we additionally compare the variants of the GOGCN with respect to the following perspectives to demonstrate its performance: (A) the effect of the operational aggregator and (B) the effect of the geographical aggregator.The following GOGCN variants are designed for comparison.

(1) GOGCN-NO: A variant of GOGCN without an operational aggregator.

(2) GOGCN-NG: A variant of GOGCN without a geographical aggregator.

The ablation study results on the testing dataset are shown in Table 3.

Effects of the operational aggregator: We compare the performance of GOGCN with GOGCN-NO using a real-world dataset to investigate the effectiveness of the operational aggregator.The results show that GOGCN performs better than GOGCN-NO, which confirms the superiority of introducing the global flight operation pattern to our model.

Effects of the geographical aggregator:We compare the performance of GOGCN with GOGCN-NG on a real-world dataset to investigate the effectiveness of the geographical aggregator.It is obvious that GOGCN achieves better performance in terms of all evaluation metrics, implying that the regional geographic interaction can provide supplementary information to benefit our model.

Difference between the effects of the two components: In Table 3,it can be observed that GOGCN-NG performs better than GOGCN-NO, which indicates the greater importance of the operational aggregator for the case study.The geographical aggregator extracts node feature information by gathering limited spatially close neighbors.Instead, based on the abundant air routes, the operational aggregator generates more accurate feature embeddings.

6.Case study of Chinese airport network

6.1.Case description

To capture the characteristics of flight delays in the Chinese airport network, we employ the pre-trained prediction model in Section 5 for practical flight delay prediction.A realworld dataset, including 1,874,591 scheduled departure flights that connect 209 civil airports between June 2018 and February 2019, is collected for the case study.The dataset contains domestic flights covering the entire day(00:00–24:00).In addition, we employ the absolute value of the difference betweenthe actual and predicted values as the prediction error (i.e.,accuracy) of each airport.

Table 3 Results of ablation study on testing dataset.

Table 2 Performance comparison of different approaches on testing dataset.

6.2.Results analysis

Tables 4 and 5 show the top 10 airports with the best and worst prediction results of the Chinese civil airports, respectively.Specifically, Ankang Fuqiang Airport (ZLAK) performs best in the Chinese airport network, while Xiamen Gaoqi International Airport (ZSAM) achieves the worst prediction results.The main reasons accounting for the abovementioned results can be attributed to the continuous thunderstorms in summer and the severe snowstorms in winter.For example, the coastal region of southeastern China is frequently disturbed by typhoons (e.g., Super Typhoon Mangkhut, Super Typhoon Kong-rey, and Super Typhoon Yutu)throughout the year.As a result, certain airports along the coast are regularly affected by heavy flight delays.In addition,northeastern China is often affected by severe snowstorms in winter, which causes massive flight cancellations or even temporary airport closures.

The prediction error for each airport is defined as the absolute value of the difference between the actual and predicted values.The figure in the left shows the prediction results of 209 Chinese airports in relation to their handling capacity.Furthermore, the prediction results of 37 ten-million-level airports are carefully analyzed in the right figure (Ten-millionlevel airport is defined as airports with over 10 million passengers per year, and the departure punctuality is defined as departure flight average punctuality of each airport in 201842).

Moreover,Fig.3(a)shows the prediction result of 209 Chinese airports in relation to their handling capacity.It is obvious that the proposed method performs better in spoke airports than in busy airports (e.g., the performance of tenmillion-level airport).Actually, the Chinese airport network exhibits a typical hub-and-spoke status leading to great differences in the absorption and diffusion of flight delays among different airports.The hub airport, representing the rare sample in the Chinese airport network,handles massive flight traffic based on superior infrastructures.However, these airports are characterized by highly fluctuating flight delays due to full-load operation.The spoke airport, occupying the vast majority of the Chinese airport network,handles limited flight traffic based on general infrastructures.And,these airports are characterized by smooth flight delays and little difficulty in delay forecasting.

Table 4 Top 10 airports with the best prediction error.

Table 5 Top 10 airports with the worst prediction error.

The prediction result of 37 ten-million-level airports is shown in Fig.3(b).It can be easily observed that the prediction error increases gradually as the departure punctuality decreases.Particularly, all airports are listed in descending order of punctuality, and there are four distinct protruding segments in the prediction error sequence.We select five representative airports from the anomalous segments to further analyze the results.

(1) Xi’an Xianyang International Airport (ZLXY): Xi’an is located in central China and has a warm, temperate,semi-humid,continental monsoon climate with pleasant weather and a low frequency of extreme weather.Moreover,ZLXY plays a hub role in the Chinese airport network and has superior departure punctuality due to its advanced infrastructure.The proposed method achieves the best performance due to a suitable climate and excellent departure punctuality.

(2) Dalian Zhoushuizi International Airport(ZYTL):Dalian is located in the Liaodong Peninsula,surrounded by the sea on three sides with sufficient water vapor.Foggy weather frequently occurs in summer, and the number of foggy days is significantly higher in June and July.Foggy weather reduces visibility and prevents flights from taking off and landing normally, which creates critical challenges to flight delay prediction.

(3) Haikou Meilan International Airport (ZJHK): Hainan Island,located in the South China Sea,suffers from severe summer tropical cyclones annually.Haikou suffers from typhoons throughout the year (e.g., Tropical Storm Ewiniar, Tropical Storm Son-Tinh, and Severe Tropical Storm Bebinca).Strong wind and rainstorms severely interfere with normal flight operation and lead to fluctuating flight delays, which decreases the performance of the proposed method.

(4) Harbin Taiping International Airport (ZYHB): Harbin,located in northeastern China, is highly susceptible to heavy snowstorms in winter due to its high latitude and proximity to the sea.Such weather conditions cause massive flight cancellations or even temporary airport closures, which decrease the performance of the proposed method.

(5) Xiamen Gaoqi International Airport(ZSAM):Xiamen is located on the subtropical coast and has a subtropical,maritime monsoon climate with frequent typhoons and abundant rainfall(e.g.,Super Typhoon Mangkhut,Severe Tropical Storm Ewiniar, and Severe Tropical Storm Barijat).Such weather conditions result in frequent flight delays or cancellations, which brings inevitable errors in flight delay prediction.

Fig.3 Illustration of prediction results of Chinese civil airports.

7.Conclusions

In this paper, a novel GOGCN is proposed for multi-airport flight delay prediction.To effectively model the regional geographic interaction in airport networks, we develop a geographical aggregator to extract the similar nature of spatially close airports.The proposed aggregator is proved to be permutation invariant, which is an important property of GCN aggregators.Additionally, we employ the weighted sum of the geographical aggregator and operational aggregator to simultaneously capture both regional geographic information and global flight operation pattern hidden in airport networks.Comprehensive experimental results based on a real dataset indicate that the proposed GOGCN achieves better performance than state-of-the-art baseline methods.

The study of using graph neural networks to investigate the multi-airport flight delay prediction problem could be extended further.For example, considering the complicated operation regulations in air transportation, it would be interesting to integrate deep learning models with operating rules of air traffic management.Moreover, a major limitation of deep learning methods is that they do not facilitate humanintelligible explanations of their predictions.It is also interesting to identify subgraph structures and small subsets of node features that play critical roles in graph-based machine learning tasks.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.71731001, U2133210, and U2033215, 61822102).