Content uploaded by Ziyu Jia
Author content
All content in this area was uploaded by Ziyu Jia on Oct 17, 2023
Content may be subject to copyright.
Delay Propagation Network in Air Transport Systems Based on
Refined Nonlinear Granger Causality
Ziyu Jiaa,b,c, Xiyang Caia, Yihan Hua, Junyu Ji a, and Zehui Jiaod
aSchool of Computer and Information Technology, Beijing Jiaotong University, Beijing,
China; bBeijing Key Laboratory of Traffic Data Analysis and Mining, Beijing, China; cKey
Laboratory of Intelligent Passenger Service of Civil Aviation, CAAC, Beijing, China;
dComputer Science & Engineering, University of California San Diego, La Jolla, CA, USA
ARTICLE HISTORY
Compiled October 17, 2023
ABSTRACT
To probe deeply into the underlying mechanism of flight delay propagation, we con-
struct the delay propagation networks among airports via refined nonlinear Granger
causality, which is more fitting on nonlinear airport delay data and obtain more pre-
cise causal estimation. Specifically, the causal analysis method is used to determine
the delay propagation relationship between airports. The delay propagation rela-
tionship of these airport pairs constitutes an airport delay propagation network, in
which the nodes and edges of the network respectively represent the specific airport
and the delay propagation relationship between airports. Complex network theory
is then applied to analyze the global structure of delay propagation networks, indi-
cating that small airports are inclined to cause or exacerbate delays. In addition, the
airlines’ delay propagation networks display that the delay propagation by large air-
lines spread wider while the delay propagation by small airlines spread faster. These
findings can be employed to design strategies for delay propagation dampening.
KEYWORDS
Granger causality; delay propagation; complex network; conditional mutual
information
1. Introduction
Flight delays seriously affect the safety and cost efficiency of air transportation systems
and gradually become a global challenge (Zhang et al. 2019; Duytschaever 1993; Cook
and Tanner 2011). Initial flight delays are attributed to a variety of reasons, such
as extreme weather and air traffic control. Usually, propagation delays occur due to
connected resources and the most common resource is the aircraft (Kafle and Zou
2016; Hao et al. 2014). For example, if the same aircraft is scheduled to fly multiple
legs, earlier delays affect subsequent missions of the same aircraft (Lan, Clarke, and
Barnhart 2006). Besides, the crews shift among diverse aircrafts, making flight delay
to disseminate to the other flights (Beatty et al. 1999; Wang et al. 2017). Hence, the
small initial delay often results in the large delay propagation (Daqing et al. 2014;
Meng and Zhou 2011).
CONTACT Ziyu Jia. Email: ziyujia@b jtu.edu.cn
In order to alleviate the large delay propagation, many researchers have made a
lot of efforts on the local dynamics of some flights, airports or airlines (Wang et al.
2020b,a; Evler et al. 2020; G¨uvercin, Ferhatosmanoglu, and Gedik 2020; Lee, Marla,
and Jacquillat 2020). For example, Beatty et al. (1999) and Wu and Law (2019) pro-
posed the idea of delay multiplier to quantify delay propagation. In addition, as the air
transport system is a complex system, some researchers have built large and complex
networks for analysis to fully grasp the characteristics of this system(Barrat et al.
2004; Guimera et al. 2005; Opsahl et al. 2008; Wuellner, Roy, and D Souza 2010;
Fleurquin, Ramasco, and Eguiluz 2013). Campanelli et al. proposed a novel method
including slot reallocation and swapping for the European aviation network (Campan-
elli et al. 2014). Afterward, they evaluated the impact of disruptions on European and
American aviation networks based on simulated delay propagation (Campanelli et al.
2016). The above work not only assists air managers to understand dynamic air trans-
portation, but also gives theoretical guides for the development of air management
strategies. However, few studies have considered the interdependence of delay time
series to study delayed propagation. Exploring the systematic framework of causal-
ity between airports is challenging. In recent years, researchers have begun to use
causal analysis to explore complex systems. The classic Granger causality test (GC)
is increasingly utilized in various fields including physiology (Kugiumtzis et al. 2017;
Kugiumtzis and Kimiskidis 2015) and finance (Papana et al. 2017b,a). Causal analy-
sis can explore interactions in dynamical systems, which helps reveal the underlying
mechanisms of complex systems.
The air transport systems are the representative complex systems. So far, some
studies have explored the delayed propagation of air transport systems through causal
analysis methods. These efforts provide a new perspective for studying the delay prop-
agation mechanism between airports. For example, using the classic linear GC method
(Granger 1969), Zanin, Belkoura, and Zhu (2017) and Du et al. (2018) studied delay
propagation by delay time series and construct delay networks between airports. Nev-
ertheless, because of the complexity of present transport system, the flight delay time
series is usually high-dimensional and non-linear. The traditional GC method is a com-
mon model to deal with linear time series (Maziarz 2015). Therefore, the information-
based nonlinear GC (NGC) can be used as an alternative method to analyze delay
time series, which is a common method for measuring the causality of nonlinear time
series (Faes, Nollo, and Porta 2011). However, the information-based nonlinear GC
produces bias in evaluating high-dimensional conditional mutual information (CMI)
(Wibral, Vicente, and Lizier 2014). Hence, it is challenging to accurately detect the
causality of multivariate time series.
To solve the above challenge, we propose refined nonlinear Granger causality
(RNGC) , which uses a low-dimensional approximation CMI (LA-CMI) in calculation
to obtain a more precise estimation of nonlinear multivariate causality. The RNGC
overcomes the curse of dimensionality by the LA-CMI. Then, its validity is verified
on the simulated multivariate time series. Besides, we use the RNGC to detect the
delay propagation of airport systems. We show the causality framework of delay prop-
agation, which relies on the interdependence of nonlinear airport delay time series.
To understand the delay interaction relationship among airports, we build the global
structure of delay propagation networks, which exhibiting the range and direction of
propagation. The results of the study indicate the importance of small airports, which
are inclined to cause or exacerbate delays. In addition, the analysis results show that
the delay propagation by large airlines spread wider while the delay propagation by
small airlines spread faster. These insights help aviation managers comprehend the
2
mechanisms of delay propagation among airports and provide theoretical support for
reducing flight delays. We will open source all data sets and codes.
The remainder of this article is designed as follows. We put forward a RNGC method
based on information theory in Section 2. In Section 3, we demonstrate the accuracy
of the RNGC method through simulation experiments. In Section 4, we describe the
data sets on the air transport system and report the results of using complex network
theory to analyze data sets. And Section 5 summarises the major conclusions.
2. Methods
Traditional nonlinear Granger causality estimates the causality of nonlinear systems
based on information theory. Generally, the traditional method uses CMI to measure
the causal relationship between time series. However, for high-dimensional time se-
ries, the estimation of CMI usually produces bias. Therefore, the calculated causality
is inaccurate. To solve this problem, we propose a low-dimensional approximation
method to estimate CMI, and then detect more accurate causality. First, the tradi-
tional nonlinear Granger causality is shown in this section. Then, the low-dimensional
approximation CMI is presented. Finally, the RNGC method is proposed.
2.1. Nonlinear Granger causality
The nonlinear Granger causality based on information theory, an extension of tradi-
tional Granger causality, has been extensively employed to estimate causality in nonlin-
ear systems. A whole dynamical system consists of Ksubsystems X, Y, Z1, . . . , ZK−2,
where Xdenotes a driving subsystem, Ydenotes a target subsystem, and Zdenote
the other subsystems. The multivariate causality from Xto Yon Zis represented
generically as:
NGCX→Y|Z=IYn;XL
n|YL
n, ZL
n
=HYn|YL
n, ZL
n−HYn|XL
n, Y L
n, ZL
n,(1)
where XL
n={Xn−1,· · · , Xn−L},YL
n={Yn−1,· · · , Yn−L}, and ZL
n=
{Zn−1,· · · , Zn−L}are variable sets that represent historical states. Ldenotes max-
imum lag range for each variable. The N GCX→Y|Zmeasures the information supplied
by the XL
nabout Yn.
The key to compute nonlinear Granger causality NGCX→Y|Zby nonuniform embed-
ding method is to construct candidate embedding vectors Vn∈M=XL
n, Y L
n, ZL
n.
We generate an empty candidate set V(0)
n=∅, and then the construction process is
introduced as follows:
•In the first step s= 1, we choose the candidate vector V(1)
n∈M, which is tested
by a criterion:
V(1)
n=W(1)
n= argmax
Wn∈M
I(Yn;Wn).(2)
•In the step s > 1, the candidate vector is expanded via adding W(s)
n, which
provides most information for Ynwhen V(s−1)
n={W(1)
n, . . . , W (s−1)
n}already
3
exists. W(s)
nis tested via a CMI criterion:
W(s)
n= argmax
Wn∈M\V(s−1)
n
IYn;Wn|V(s−1)
n.(3)
The choice loop is stopped when W(s)
ndoes not meet significance test. Then, this
candidate set is obtained as Vn=V(s−1)
n.
•The NGC measure is computed as:
NGCX→Y|Z=HYn|VY
n, V Z
n−H(Yn|Vn).(4)
where Vn={VX
n, V Y
n, V Z
n},VX
n, V Y
nand VZ
n∈Vncome from variables X, Y, and
Z, respectively.
2.2. The Proposed Method
2.2.1. The motivation for applying the LA-CMI
The formation process of candidate vector is regarded as the feature selection process
based on information theory (Han et al. 2018). The selected candidate vector provides
the most information for the target variable and minimizes redundancy information
between the selected variable set. Therefore, each step needs to estimate the accu-
rate CMI and select the best variable. Nevertheless, as candidate vector dimension
increases, the curse of dimensionality appears due to the increase of state space di-
mension, which leads to the gradually inaccurate estimation of CMI (Wibral, Vicente,
and Lizier 2014; Jia et al. 2019). To tackle the issue, we propose a LA-CMI to improve
the NGC method in the following sections.
2.2.2. Low-dimensional approximation for CMI
A major problem for feature selection research is how to accurately estimate the high-
dimensional CMI. The researchers have done a lot of excellent work. For example, the
JMI criterion (Yang and Moody 2000), the MRMR criterion (Peng, Long, and Ding
2005), and the DISR criterion (Meyer and Bontempi 2006). Moreover, Brown et al.
(2012) proposed a parameterized general criterion based on the above criteria:
J(Ws) = I(Ws;Yn)−ϵX
Wj∈V
I(Ws;Wj) + ηX
Wj∈V
I(Ws;Wj|Yn) (5)
where the different criteria relay on the values of ϵand η, such as the JMI criterion
is gained when ϵ=η= 1/|V|; the MRMR is gained when ϵ= 1/|V|,η= 0. Fur-
thermore, Vinh et al. (2016) proposed the RelaxMRMR to improve the traditional
criteria. Specifically, higher-order feature interplays are studied within this RelaxM-
RMR by relaxing specific conditional independence assumptions.
The innovativeness of RelaxMRMR compared to Equation (5) is the I(Ws;Wi|Wj),
4
which contains the second-order interplays between features.
J(Ws) = I(Ws;Yn)−ϵX
Wj∈V
I(Ws;Wj)+ηX
Wj∈V
I(Ws;Wj|Yn)−θX
Wi,Wj∈V;i=j
I(Ws;Wi|Wj),
(6)
where ϵ=η= 1/|V|and θ= 1/|V|(|V| − 1).
In our work, the above RelaxMRMR criterion in Equation (6) is utilized as the
replacement for high-dimensional CMI. We define the target variable Ynand candidate
vector V=W1, W2,...W(s−1), an expression for low-dimensional approximation of
CMI is as follows:
Wn=argmax
Ws∈M\V
I(Ws;Yn)−1
|V|X
Wj∈V
I(Ws;Wj) + 1
|V|X
Wj∈V
I(Ws;Wj|Yn)
−1
|V|(|V| − 1) X
Wi,Wj∈V;i=j
I(Ws;Wi|Wj)
(7)
2.2.3. LA-CMI for nonlinear Granger causality
We present the Refined-NGC (RNGC) method, which utilizes the LA-CMI criterion
when the candidate vector dimension is high. We set a criterion control factor mto
determine when the low-dimensional approximate CMI criterion is used. We generate
an empty candidate set V(0)
n=∅. Then, the details of the RNGC method are as
follows.
•In the first step s= 1, we choose the candidate vector V(1)
n∈M, which is tested
by a criterion :
V(1)
n=W(1)
n= argmax
Wn∈M
I(Yn;Wn).(8)
•In the step 1 < s ≤m, the candidate vector is expanded via adding W(s)
n, which
provides most information for Ynwhen V(s−1)
n={W(1)
n, . . . , W (s−1)
n}already
exists. W(s)
nis tested via a standard CMI criterion (Equation 3).
•In the step s>m, the low-dimensional approximation CMI criterion (Equation
7) is used for embedding.
•The choice loop is stopped when W(s)
ndoes not meet significance test. Then, this
candidate set is obtained as Vn=V(s−1)
n.
•The proposed RNGC measure:
RNGCX→Y|Z=HYn|VY
n, V Z
n−H(Yn|Vn).(9)
where Vn={VX
n, V Y
n, V Z
n},VX
n, V Y
nand VZ
n∈Vncome from variables X, Y, and
Z, respectively.
5
3. Simulation study
In this section, we employ simulation experiments to validate the effectiveness of
RNGC method. The proposed RNGC method is compared to three traditional causal-
ity methods (GC , TE, and NGC) on the Henon maps. To guarantee the robustness
of the experiment, 100 realizations are generated for the Henon maps. Each causal
strength obtained is the average of 100 realizations. In addition, the significance level
αis set to 0.05 in the significance test for all methods (Kugiumtzis 2013).
X1
X2
X3XK-1
Xk
Figure 1. The true causality network of this Kcoupled Henon maps.
We define the Kcoupled Henon maps (Kugiumtzis 2013):
xi,t =1.4−x2
i,t−1+ 0.3xi,t−2for i= 1, K
1.4−(0.5C(xi−1,t−1+xi+1,t−1) + (1 −C)xi,t−1)2+ 0.3xi,t−2for i= 2, . . . , K −1
(10)
where Cdenotes coupling strength. In addition, true causality of this coupled Henon
maps is shown in Figure. 1. Table 1 presents three evaluation metrics obtained on this
K= 5 coupled Henon maps for varying sequence length n. The results present that the
RNGC measure performs better than other methods. Moreover, the proposed RNGC
measure is greater than 0.9 in all evaluation metrics. As the time series changes, the F1
score of the proposed RNGC measure does not change much. Therefore, the proposed
RNGC measure is less influenced by time series length and maintains a high accuracy
to detect coupling. To study the influence of different Kon causal detection, we fix
n= 256 and change the variable Kto get the evaluation metrics as shown in Table
2. As the number of variables increases, the accuracy of detecting coupling decreases.
However, the proposed RNGC method still performs better than other methods in
terms of various evaluation metrics. In addition, the results indicate that the GC
method is not accurate because the GC method may be more suitable to linear time
series. Therefore, the RNGC method is more fitting to process nonlinear delay time
series.
4. Case study of air transport system
In the section, we first introduce the air transport system data sets. We then construct
the airport delay time series and show the nonlinearity of the delay time series. Finally,
we construct and investigate the delay propagation networks.
6
Table 1. The evaluation metrics from the coupled Henon maps with varying length
n. The experimental parameters are α= 0.05, K= 5, and C= 0.9.
Sensitivity Specificity F1 score
n= 256
GC (Granger 1969) 0.901 0.937 0.859
TE (Montalto, Faes, and Marinazzo 2014) 0.993 0.957 0.934
NGC (Faes, Nollo, and Porta 2011) 0.985 0.960 0.938
RNGC 0.988 0.964 0.941
n= 512
GC (Granger 1969) 0.993 0.887 0.845
TE (Montalto, Faes, and Marinazzo 2014) 1.000 0.942 0.916
NGC (Faes, Nollo, and Porta 2011) 1.000 0.944 0.919
RNGC 1.000 0.952 0.930
n= 1024
GC (Granger 1969) 1.000 0.866 0.825
TE (Montalto, Faes, and Marinazzo 2014) 1.000 0.902 0.865
NGC (Faes, Nollo, and Porta 2011) 1.000 0.913 0.878
RNGC 1.000 0.944 0.919
Table 2. The evaluation metrics from the coupled Henon maps with varying K.
The experimental parameters are α= 0.05, n= 256, and C= 0.9.
Sensitivity Specificity F1 score
K= 5
GC (Granger 1969) 0.901 0.937 0.859
TE (Montalto, Faes, and Marinazzo 2014) 0.993 0.957 0.934
NGC (Faes, Nollo, and Porta 2011) 0.985 0.960 0.938
RNGC 0.988 0.964 0.941
K= 10
GC (Granger 1969) 0.911 0.904 0.755
TE (Montalto, Faes, and Marinazzo 2014) 0.978 0.945 0.864
NGC (Faes, Nollo, and Porta 2011) 0.979 0.947 0.869
RNGC 0.980 0.950 0.875
K= 15
GC (Granger 1969) 0.936 0.909 0.711
TE (Montalto, Faes, and Marinazzo 2014) 0.965 0.959 0.847
NGC (Faes, Nollo, and Porta 2011) 0.966 0.961 0.854
RNGC 0.967 0.963 0.861
4.1. Data description
The dataset is given by Civil Aviation Administration of China (CAAC), compris-
ing flight status information in China from 1 to 31 December 2016. In addition, the
dataset includes 313,477 flights, 205 airports, and 39 airlines. The information avail-
able for each flight is the aircraft registration number, scheduled departure time, actual
departure time, scheduled arrival time, actual arrival time, airline International Air
Transport Association (IATA) code, departure airport IATA code, and arrival airport
IATA code.
Table 3. Information for each flight.
Available information for each flight
Aircraft registration number Actual arrival time
Scheduled departure time Airline IATA
Actual departure time Departure airport IATA
Scheduled arrival time Arrival airport IATA
7
4.2. Time series preprocessing and analysis
To capture the delayed disseminate among airports, we construct the arrival delay
time series (DTS) for all airports, which describes on-time performance concerning all
airports. Each time point in the formed DTS denotes average flight delay time per
unit hour. Specifically, the average DTS for i-th airport is defined as:
Di(p, q) = P(Ti
a−Ti
s)
Ni(p, q),1≤p≤31,0≤q≤23,(11)
where Di(p, q) is average delay time for i-th airport on day pand hour q. And Ti
aand
Ti
sare the actual and scheduled flight arrival time, respectively. Ni(p, q) is the number
of flights for the i-th airport per hour. To confirm the nonlinearity of DTS, we employ
the Lyapunov exponent. Generally speaking, a dynamic system can be regarded as a
nonlinear system when the largest Lyapunov exponent is positive (Wolf et al. 1985).
The largest Lyapunov exponent of the data is 0.15, which indicates that the flight
delay system exhibits significant nonlinearity.
4.3. Network metrics
In this study, complex network is used to investigate the causality of delayed propa-
gation in air transport systems.
Degree. The degree of each airport represents the number of remaining airports with
which it has a delayed propagation relationship. The in-degree of airport iindicates
the number of remaining airports affecting airport i. Conversely, the out-degree is
obtained. The network average degree denotes each airport affects others on average.
Assortativity coefficient (Newman 2002). A positive value of assortativity coefficient
indicates that the network is an assortative network. In the assortative network, the
airports prefer to link preferentially toward other airports with the similar degree.
Conversely, it is called disassortative network.
Link density. The larger the link density value, the more the number of network
links. That is to say, when the link density value is large, delays are easily propagated
through the network.
Average clustering coefficient (Watts and Strogatz 1998). The average aggregation
coefficients is computed as fellow after getting local aggregation coefficients for all
nodes. And the larger the value of average aggregation coefficient, the higher degree
of aggregation for airport nodes within this network.
Modularity (Newman 2004). The modularity can measure the strength of the net-
work community structure. The entire network is made up of several communities.
The spread of delays among airports within each community is relatively dense. In
contrast, the spread of delays among communities is relatively sparse (Arenas et al.
2007). The higher modularity, the higher regionalization degree of delay propagation.
Efficiency (Latora and Marchiori 2001). The efficiency of the network reflects the
ease of delay propagation between two airports. The greater the value of efficiency,
the faster the delay propagate to entire network.
8
4.4. Analysis of air transport system
4.4.1. Basic characteristics for delay propagation network
The basic characteristics for delay propagation network constructed by using delay
time series are analyzed. The proposed RNGC method is utilized to construct a delayed
propagation network, which contains 205 airport nodes as shown in Figure. 2. The
larger the node’s degree of network, the larger the nodes. Conversely, the smaller the
node’s degree of network, the smaller the nodes. Figure. 2 presents some small airports
with larger nodes, such as GMQ and XYI.
Figure 2. The delay propagation network between airports. The links denote delay propagation between
airports. The degree of a node is proportional to its size.
0 1 0 2 0 3 0 4 0
0
2 0
4 0
6 0
8 0
PEK
C A N
N L H
G M Q
XYI
O utdegree
N u m b e r o f fl i g h t s ( 1 E 3 )
(a) Out-degree and the number of flights
for each airport.
0 1 0 2 0 3 0 4 0
0
2
4
6
PEK
C A N
W N Z
L H W
Indegree
N u m b e r o f fl i g h t s ( 1 E 3 )
(b) In-degree and the number of flights for
each airport.
Figure 3. The relationship between degree and the number of flights at the airports. GMQ, NLH, and XYI
are small airports. PEK and CAN are large airports.
Figure. 3 further reflects the relationship between flight number and airport node’s
degree, where Figure. 3(a) shows the relationship between out-degrees of nodes and
the number of flights are inversely related; that is to say, the node’s degree with a
larger number of flights is small, and it is consistent with other air transport networks
(Cook, Tanner, and Zanin 2013; AhmadBeygi et al. 2008; Xiao et al. 2020; Jetzki 2009;
Jia et al. 2020). Figure. 3(b) reflects the relationship between in-degrees of nodes and
9
the number of flights and it is obvious that there is no obvious relationship between
them. However, most of the in-degrees of nodes with a larger number of flights are still
small.
Table 4. Delay propagation network topological metric
Topological metric Delay propagation relational
network metrics value
Average value of 1000 random
network metrics
Average degree 2.62 2.62
Link density 0.01 0.01
Assortativity coefficient -0.09 -0.01
Average clustering coefficient 0.02 0.01
Modularity 0.63 0.42
Efficiency 0.39 0.32
In summary, airports with a small number of flights in the delay propagation network
often experience delays. These airports are usually small airports, and the aviation
resources and mitigation measures of these small airports may not be as good as those
of large airports. Hence, small airports may increase the spread of delays.
Table 4 shows the network topological indicators calculated from the delay prop-
agation network and the average values of 1000 random networks constructed from
the delay propagation network. From the value of the average degree 2.62, it is known
that each airport spreads delays to multiple other airports. The value of the link
density is 0.01 indicating that the delay propagation network is sparse. The average
clustering coefficient, modularity, and efficiency of the network are higher than those
of the random networks, which indicates that the delay propagation network has a
strong clustering trend, relatively obvious community structure. In other words, there
are some sub-networks (network communities) with more internal connections, but
the connections between sub-networks are relatively sparse. In fact, the community
structure is in the constructed delay propagation network. Therefore, airports in each
network community may easily spread delays to each other. However, the assortativity
coefficient of the delay propagation network is lower than that of the random networks,
which shows that the network has higher disassortative. That is, the nodes with the
larger degree tend to link those with smaller degree in the network. Because the de-
gree of nodes is inversely related to the number of flights, there is delay propagation
between airports with more flights and airports with fewer flights.
4.4.2. Airline characteristics for delay propagation network
To investigate the differences in delay propagation for each airline, the airline’s monthly
delay propagation network is constructed in this section. The constructed airline net-
works are equivalent to stratifying the delay propagation network used in the section
4.4.1 according to the airlines. All airlines are ordered by flight number and the top
10 airlines are selected. The IATA codes are CZ, MU, CA, HU, ZH, MF, SC, 3U, GS,
and FM respectively.
Figure.4 shows that 3U airline has fewer flights, but total delay time is similar to
that of CA airline. Figure.5 shows the number of nodes and links for each airline’s
delay propagation network. It is obvious that the number of links and nodes for three
large airline networks is greater than those of remaining small airlines. Because the
three major airlines involve more cities and flights, the delay spread is larger.
The efficiency and assortativity coefficient for each airline’s delay propagation net-
work are illustrated in Figure.6. Figure.6(a) presents the efficiency of delay propagation
networks, which indicates that the delay propagation of small airlines is more efficient.
10
C Z M U C A H U Z H M F S C 3 U G S F M
0
1 0
2 0
3 0
4 0
5 0
N u m b e r o f f l i g h t s ( 1 E 3 )
I A T A c o d e f o r a i r l i n e s
N u m b e r o f f l i g h t s
S u m m a r y o f a r r i v a l d e l a y s
0
1 0
2 0
3 0
S u m m a r y o f a r r i v a l d e l a y s ( 1 E 4 m i n )
Figure 4. The sum of the number of flights and delay time for each airline in December 2016.
(a) The number of nodes of each airline’s delay prop-
agation network.
(b) The number of links of each airline’s delay prop-
agation network.
Figure 5. The number of nodes and links of each airline’s delay propagation network.
However, the delay propagation of large airlines is inefficient, because the network is
large or there are resources to mitigate delays. As shown in Figure.6(b), Each small
airline usually has a small assortativity coefficient, especially MF, SC and GS airlines.
And the efficiency of delay propagation networks are improved in small airlines.
5. Conclusion
In the research, we propose a refined causal analysis approach to reveal the delay
propagation among airports. The proposed RNGC method overcomes the curse of
dimensionality for the traditional method by using the low-dimensional approximation
CMI, and accurately detects the causality of simulated time series. In addition, in
actual applications, the delay propagation networks are established based on the DTS
of each airport. Then, network analysis metrics are utilized to reflect the macroscopic
presentation of delayed propagation.
Understanding the mechanism of delayed propagation can assist managers to de-
velop aviation planning. We adopt complex network theory to investigate this mech-
anism. The results of the analysis emphasize the importance of small airports, which
are inclined to cause or exacerbate delays. In addition, the analysis results also show
11
(a) The efficiency of each airline’s delay propagation
network.
(b) The assortativity coefficient of each airline’s delay
propagation network.
Figure 6. The efficiency and assortativity coefficient of each airline’s delay propagation network.
that the delay propagation by large airlines spread wider while the delay propagation
by small airlines spread faster. Therefore, it is necessary to consider the characteristics
of the airline’s delay propagation when there is an emergency of flight delays.
These insights are employed to develop air traffic schemes to mitigate or suppress
the delayed spread. The research based on delay propagation networks to detect de-
layed propagation can be extended. For instance, the delay propagation networks con-
structed in this study are unweighted networks. In the future, the edge weights can be
considered because the edge weights reflect the degree of causality.
Disclosure statement
No potential conflict of interest was reported by the authors.
Acknowledgment
This project was funded by the Fundamental Research Funds for the Central Univer-
sities (Grant No.2020YJS025) and partially supported by the Zhejiang Lab’s Interna-
tional Talent Fund for Young Professionals.
References
AhmadBeygi, Shervin, Amy Cohn, Yihan Guan, and Peter Belobaba. 2008. “Analysis of the
potential for delay propagation in passenger airline networks.” Journal of air transport
management 14 (5): 221–236.
Arenas, Alex, Jordi Duch, Alberto Fern´andez, and Sergio G´omez. 2007. “Size reduction of
complex networks preserving modularity.” New Journal of Physics 9 (6): 176.
Barrat, Alain, Marc Barthelemy, Romualdo Pastor-Satorras, and Alessandro Vespignani. 2004.
“The architecture of complex weighted networks.” Proceedings of the national academy of
sciences 101 (11): 3747–3752.
Beatty, Roger, Rose Hsu, Lee Berry, and James Rome. 1999. “Preliminary evaluation of flight
delay propagation through an airline schedule.” Air Traffic Control Quarterly 7 (4): 259–270.
Brown, Gavin, Adam Pocock, Ming-Jie Zhao, and Mikel Luj´an. 2012. “Conditional likelihood
12
maximisation: a unifying framework for information theoretic feature selection.” Journal of
machine learning research 13 (Jan): 27–66.
Campanelli, B, P Fleurquin, VM Egu´ıluz, JJ Ramasco, A Arranz, I Etxebarria, and C Ciruelos.
2014. “Modeling reactionary delays in the European air transport network.” Proceedings of
the Fourth SESAR Innovation Days, Schaefer D (Ed.), Madrid .
Campanelli, Bruno, Pablo Fleurquin, Andr´es Arranz, Izaro Etxebarria, Carla Ciruelos,
V´ıctor M Egu´ıluz, and Jos´e J Ramasco. 2016. “Comparing the modeling of delay propaga-
tion in the US and European air traffic networks.” Journal of Air Transport Management
56: 12–18.
Cook, Andrew, Graham Tanner, and Massimiliano Zanin. 2013. “Towards superior air trans-
port performance metrics–imperatives and methods.” Journal of Aerospace Operations 2
(1-2): 3–19.
Cook, Andrew J, and Graham Tanner. 2011. “European airline delay cost reference values.” .
Daqing, Li, Jiang Yinan, Kang Rui, and Shlomo Havlin. 2014. “Spatial correlation analysis of
cascading failures: congestions and blackouts.” Scientific reports 4: 5381.
Du, Wen-Bo, Ming-Yuan Zhang, Yu Zhang, Xian-Bin Cao, and Jun Zhang. 2018. “Delay
causality network in air transport systems.” Transportation research part E: logistics and
transportation review 118: 466–476.
Duytschaever, DIRK. 1993. “The development and implementation of the eurocontrol central
air traffic flow management unit (cfmu).” The Journal of Navigation 46 (3): 343–352.
Evler, Jan, Michael Schultz, Hartmut Fricke, and AJ Cook. 2020. “Development of stochastic
delay cost functions.” 10th SESAR Innovation Days .
Faes, Luca, Giandomenico Nollo, and Alberto Porta. 2011. “Information-based detection of
nonlinear Granger causality in multivariate processes via a nonuniform embedding tech-
nique.” Physical Review E 83 (5): 051112.
Fleurquin, Pablo, Jos´e J Ramasco, and Victor M Eguiluz. 2013. “Systemic delay propagation
in the US airport network.” Scientific reports 3: 1159.
Granger, Clive WJ. 1969. “Investigating causal relations by econometric models and cross-
spectral methods.” Econometrica: Journal of the Econometric Society 424–438.
Guimera, Roger, Stefano Mossa, Adrian Turtschi, and LA Nunes Amaral. 2005. “The world-
wide air transportation network: Anomalous centrality, community structure, and cities’
global roles.” Proceedings of the National Academy of Sciences 102 (22): 7794–7799.
G¨uvercin, Mehmet, Nilgun Ferhatosmanoglu, and Bugra Gedik. 2020. “Forecasting Flight De-
lays Using Clustered Models Based on Airport Networks.” IEEE Transactions on Intelligent
Transportation Systems .
Han, Min, Weijie Ren, Meiling Xu, and Tie Qiu. 2018. “Nonuniform state space reconstruction
for multivariate chaotic time series.” IEEE transactions on cybernetics 49 (5): 1885–1895.
Hao, Lu, Mark Hansen, Yu Zhang, and Joseph Post. 2014. “New York, New York: Two ways
of estimating the delay impact of New York airports.” Transportation Research Part E:
Logistics and Transportation Review 70: 245–260.
Jetzki, Martina. 2009. “The propagation of air transport delays in Europe.” Master’s thesis,
RWTH Aachen University, Airport and Air Transportation Research .
Jia, Ziyu, Youfang Lin, Zehui Jiao, Yan Ma, and Jing Wang. 2019. “Detecting causality in
multivariate time series via non-uniform embedding.” Entropy 21 (12): 1233.
Jia, Ziyu, Youfang Lin, Yunxiao Liu, Zehui Jiao, and Jing Wang. 2020. “Refined nonuniform
embedding for coupling detection in multivariate time series.” Physical Review E 101 (6):
062113.
Kafle, Nabin, and Bo Zou. 2016. “Modeling flight delay propagation: A new analytical-
econometric approach.” Transportation Research Part B: Methodological 93: 520–542.
Kugiumtzis, Dimitris. 2013. “Direct-coupling information measure from nonuniform embed-
ding.” Physical Review E 87 (6): 062918.
Kugiumtzis, Dimitris, and Vasilios K Kimiskidis. 2015. “Direct causal networks for the study
of transcranial magnetic stimulation effects on focal epileptiform discharges.” International
journal of neural systems 25 (05): 1550006.
13
Kugiumtzis, Dimitris, Christos Koutlis, Alkiviadis Tsimpiris, and Vasilios K Kimiskidis. 2017.
“Dynamics of epileptiform discharges induced by transcranial magnetic stimulation in ge-
netic generalized epilepsy.” International journal of neural systems 27 (07): 1750037.
Lan, Shan, John-Paul Clarke, and Cynthia Barnhart. 2006. “Planning for robust airline op-
erations: Optimizing aircraft routings and flight departure times to minimize passenger
disruptions.” Transportation science 40 (1): 15–28.
Latora, Vito, and Massimo Marchiori. 2001. “Efficient behavior of small-world networks.”
Physical review letters 87 (19): 198701.
Lee, Jane, Lavanya Marla, and Alexandre Jacquillat. 2020. “Dynamic disruption management
in airline networks under airport operating uncertainty.” Transportation Science 54 (4):
973–997.
Maziarz, Mariusz. 2015. “A review of the Granger-causality fallacy.” The journal of philosoph-
ical economics: Reflections on economic and social issues 8 (2): 86–105.
Meng, Lingyun, and Xuesong Zhou. 2011. “Robust single-track train dispatching model under
a dynamic and stochastic environment: a scenario-based rolling horizon solution approach.”
Transportation Research Part B: Methodological 45 (7): 1080–1102.
Meyer, Patrick E, and Gianluca Bontempi. 2006. “On the use of variable complementarity
for feature selection in cancer classification.” In Workshops on applications of evolutionary
computation, 91–102. Springer.
Montalto, Alessandro, Luca Faes, and Daniele Marinazzo. 2014. “MuTE: a MATLAB toolbox
to compare established and novel estimators of the multivariate transfer entropy.” PloS one
9 (10): e109462.
Newman, Mark EJ. 2002. “Assortative mixing in networks.” Physical review letters 89 (20):
208701.
Newman, Mark EJ. 2004. “Fast algorithm for detecting community structure in networks.”
Physical review E 69 (6): 066133.
Opsahl, Tore, Vittoria Colizza, Pietro Panzarasa, and Jose J Ramasco. 2008. “Prominence and
control: the weighted rich-club effect.” Physical review letters 101 (16): 168702.
Papana, Angeliki, Catherine Kyrtsou, Dimitris Kugiumtzis, and Cees Diks. 2017a. “Assessment
of resampling methods for causality testing: A note on the US inflation behavior.” PloS one
12 (7): e0180852.
Papana, Angeliki, Catherine Kyrtsou, Dimitris Kugiumtzis, and Cees Diks. 2017b. “Financial
networks based on Granger causality: A case study.” Physica A: Statistical Mechanics and
its Applications 482: 65–73.
Peng, Hanchuan, Fuhui Long, and Chris Ding. 2005. “Feature selection based on mutual in-
formation: criteria of max-dependency, max-relevance, and min-redundancy.” IEEE Trans-
actions on Pattern Analysis and Machine Intelligence (8): 1226–1238.
Vinh, Nguyen Xuan, Shuo Zhou, Jeffrey Chan, and James Bailey. 2016. “Can high-order
dependencies improve mutual information based feature selection?” Pattern Recognition 53:
46–58.
Wang, Yan-Jun, Ya-Kun Cao, Chen-Ping Zhu, Fan Wu, Ming-Hua Hu, Baruch Barzel, and
HE Stanley. 2017. “Characterizing departure delays of flights in passenger aviation network
of United States.” arXiv preprint arXiv:1701.05556 .
Wang, Yanjun, Yakun Cao, Chenping Zhu, Fan Wu, Minghua Hu, Vu Duong, Michael Watkins,
Baruch Barzel, and H Eugene Stanley. 2020a. “Universal patterns in passenger flight depar-
ture delays.” Scientific reports 10 (1): 1–10.
Wang, Yanjun, Hongfeng Zheng, Fan Wu, Jun Chen, and Mark Hansen. 2020b. “A Com-
parative Study on Flight Delay Networks of the USA and China.” Journal of Advanced
Transportation 2020.
Watts, Duncan J, and Steven H Strogatz. 1998. “Collective dynamics of ‘small-
world’networks.” nature 393 (6684): 440.
Wibral, Michael, Raul Vicente, and Joseph T Lizier. 2014. Directed information measures in
neuroscience. Springer.
Wolf, Alan, Jack B Swift, Harry L Swinney, and John A Vastano. 1985. “Determining Lyapunov
14
exponents from a time series.” Physica D: Nonlinear Phenomena 16 (3): 285–317.
Wu, Cheng-Lung, and Kristie Law. 2019. “Modelling the delay propagation effects of multiple
resource connections in an airline network using a Bayesian network model.” Transportation
Research Part E: Logistics and Transportation Review 122: 62–77.
Wuellner, Daniel R, Soumen Roy, and Raissa M D Souza. 2010. “Resilience and rewiring of
the passenger airline networks in the United States.” Physical Review E 82 (5): 056101.
Xiao, Yinhong, Yaoshuai Zhao, Ge Wu, and Yizhen Jing. 2020. “Study on delay propagation
relations among airports based on transfer entropy.” IEEE Access 8: 97103–97113.
Yang, Howard Hua, and John Moody. 2000. “Data visualization and feature selection: New
algorithms for nongaussian data.” In Advances in neural information processing systems,
687–693.
Zanin, Massimiliano, Seddik Belkoura, and Yanbo Zhu. 2017. “Network analysis of Chinese
air transport delay propagation.” Chinese Journal of Aeronautics 30 (2): 491–499.
Zhang, Mingyuan, Xuting Zhou, Yu Zhang, Lijun Sun, Ming Dun, Wenbo Du, and Xianbin
Cao. 2019. “Propagation Index on Airport Delays.” Transportation Research Record 2673
(8): 536–543. https://doi.org/10.1177/0361198119844240.
15
