Characterizing the Structure of the Railway Network in China: A Complex Weighted Network Approach

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DOI: 10.1155/2019/3928260
Cite this publication
Abstract
Understanding the structure of the Chinese railway network (CRN) is crucial for maintaining its efficiency and planning its future development. To advance our knowledge of CRN, we modeled CRN as a complex weighted network and explored the structural characteristics of the network via statistical evaluations and spatial analysis. Our results show CRN as a small-world network whose train flow obeys power-law decaying, demonstrating that CRN is a mature transportation infrastructure with a scale-free structure. CRN also shows significant spatial heterogeneity and hierarchy in its regionally uneven train flow distribution. We then examined the nodal centralities of CRN using four topological measures: degree, strength, betweenness, and closeness. Nodal degree is positively correlated with strength, betweenness, and closeness. Unlike the common feature of a scale-free network, the most connected nodes in CRN are not necessarily the most central due to underlying geographical, political, and socioeconomic factors. We proposed an integrated measure based on the four centrality measures to identify the global role of each node and the multilayer structure of CRN and confirm that stable connections hold between different layers of CRN.
Research Article
Characterizing the Structure of the Railway Network in China: A
Complex Weighted Network Approach
Weiwei Cao,1Xiangnan Feng ,1Jianmin Jia,2and Hong Zhang3
1School of Economic and Management, Southwest Jiaotong University, Chengdu 610031, China
2School of Economic and Management, e Chinese University of Hong Kong (Shenzhen), Shenzhen 518172, China
3Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu 611756, China
Correspondence should be addressed to Xiangnan Feng; fengxiangnan@gmail.com
Received 23 June 2018; Revised 18 November 2018; Accepted 4 December 2018; Published 3 February 2019
Academic Editor: Roıo de O˜na
Copyright ©  Weiwei Cao et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Understanding the structure of the Chinese railway network (CRN) is crucial for maintaining its eciency and planning its future
development. To advance our knowledge of CRN, we modeled CRN as a complex weighted network and explored the structural
characteristics of the network via statistical evaluations and spatial analysis. Our results show CRN as a small-world network
whose train ow obeys power-law decaying, demonstrating that CRN is a mature transportation infrastructure with a scale-free
structure. CRN also shows signicant spatial heterogeneity and hierarchy in its regionally uneven train ow distribution. We then
examined the nodal centralities of CRN using four topological measures: degree, strength, betweenness, and closeness. Nodal
degree is positively correlated with strength, betweenness, and closeness. Unlike the common feature of a scale-free network, the
most connected nodes in CRN are not necessarily the most central due to underlying geographical, political, and socioeconomic
factors. We proposed an integrated measure based on the four centrality measures to identify the global role of each node and the
multilayer structure of CRN and conrm that stable connections hold between dierent layers of CRN.
1. Introduction
Transportation networks have enormous impacts on national
economic and social activities [, ]. As argued by Alderighi
et al. [], the structure of a network shapes operational
strategy and service quality of the system. us, it is crucial
to examine the structural characteristics of transportation
infrastructures. Complex network theory has been widely
used to analyze the structural properties of various real-world
transportation networks, including airport networks [–],
shipping networks [], subway networks [], bus networks
[, ], and railway networks [–].
Like other transportation systems, railway networks are
closely linked to sustainable regional development []. Par-
ticular attention has been paid to the topological properties
of railway networks. Sen et al. [] rst employed complex
network theory to study India’s railway network, revealing
the systems small-world character. Soh et al. [] showed that
Singapore’s railway network is almost fully connected, and its
hub nodes experience disproportionately heavy trac. e
Chinese railway network (CRN) is one of the largest railway
networks in the world and signicantly contributes to the
country’s development []. However, the structural perfor-
mance of CRN has not been given sucient attention in
authoritative journals; previous research has focused mainly
on the structural properties of CRN through pure statistical
analyses of stations’ connections [, –] while generally
overlooking the spatial properties of CRN.
During the last decade, China’s railway infrastructure
experienced a construction boom, increasing the operating
mileage of CRN dramatically from , km in  to
,  km in . us, the structure of CRN may have
changed substantially and thus needs a thorough reexamina-
tion. In this paper, we modeled CRN as a weighted network
and employed measures from complex network theory to
explore the structural characteristics of CRN. Along with
its statistical properties, we also evaluated CRN’s spatial
heterogeneity and hierarchy. Additionally, because a single
Hindawi
Journal of Advanced Transportation
Volume 2019, Article ID 3928260, 10 pages
https://doi.org/10.1155/2019/3928260
Journal of Advanced Transportation
centrality measure fails to capture the overall importance
of a node in the railway network, we proposed a data-
driven integrated measure based on the four centrality
measures (degree, strength, betweenness, and closeness) to
comprehensively quantify the importance of each node. is
measure provides meaningful insights into the national roles
of cities in CRN, which helps reveal the multilayer structure of
CRN.
e rest of this paper is structured as follows: Section
introduces data and structure measures. Section  reports
the statistical and spatial properties of CRN at a global scale.
Section  presents the centrality measures of nodes and
explores their relationships. Section  proposes an integrated
measure to reveal the role of each node in CRN. Section 
concludes the paper.
2. Data Sources and Network
Structure Measures
2.1. Data Sources. e data set analyzed was provided by
a railway bureau of China. It includes information on all
railway stations and over , domestic scheduled passenger
trains in China in . For transportation networks, a
widely used methodology for abstracting a system into a
complex network is the P-space method [], illustrated in
Figure . e P-space representation contains railway stations
as nodes and shows a connection between two nodes if a
train connects the station pair. For each connection, multiple
trains are possible, and the weight of the connection (line)
is the total number of trains between the pair of nodes.
Following previous studies [], we treated cities instead of
railway stations as nodes. All railway stations in the same
city are attributed to the city. For example, Beijing railway
station, Beijing West railway station, and Beijing South
railway station located in Beijing are all assigned to the
node for Beijing City. If a train connects any one station in
Beijing to any one station in another city, this is considered
a connection between Beijing and that city. As a result, the
network CRN has = 1192 nodes (i.e., cities) and ,
edges.
2.2. Network Structure Measures. We applied a variety
of topological measures to explore the structural char-
acteristics of CRN. e rst two measures summarized
the global scale properties of CRN, and the follow-
ing four measures characterized nodes’ centralities in the
network.
2.2.1. Average Path Length. Average path length [] is
dened as the average value of the shortest path lengths
between all node pairs in a network:
L=1
(−1
)
𝑖 ̸=𝑗𝑖𝑗 ()
where is the number of nodes and 𝑖𝑗 is the shortest path
length between node and node .
2.2.2. Clustering Coecient. e clustering coecient [] of
node is as follows:
𝑖=2𝑖
𝑖𝑖−1()
where 𝑖is the number of neighbors connected to node ,𝑖
is the actual number of edges connecting the 𝑖neighbors,
and 𝑖(𝑖)/ is the largest possible number of edges
between these neighbors.
2.2.3. Degree Centrality. e degree of a node [] is dened
as the number of neighbors in the network connected to that
node. It is represented as follows:
𝐷()=
𝑗∈𝑉𝑖𝑗 ()
where represents the set of all nodes in the network except
node ,and𝑖𝑗 is dened as  if there is a connection between
nodes and ,andotherwise.
2.2.4. Strength Centrality. As an extension of degree, strength
centrality combines the connectivity and train ow informa-
tion of a node []. Strength is formalized as follows:
𝑆()=
𝑗∈𝑉𝑖𝑗 ()
where 𝑖𝑗 represents the weight of the edge between nodes
and .
2.2.5. Betweenness Centrality. Betweenness centrality [] is
represented as follows.
𝐵(i)=
𝑗 ̸=𝑘
𝑗𝑘 ()
𝑗𝑘 ()
Here, 𝑗𝑘 counts the number of possible shortest paths
between nodes and ,𝑗() denotes the number of shortest
paths between nodes and that pass through node ,and
𝑗𝑘()/𝑗𝑘 represents the proportion of the shortest paths
between nodes and that pass through node .
2.2.6. Closeness Centrality. e closeness centrality [] of
node is represented as follows:
𝐶()=−1
𝑗∈𝑉 𝑖𝑗 ()
where is the number of nodes in the network, represents
the set of all nodes in the network except node ,and𝑖𝑗 is the
shortest path length between nodes and .
3. The Statistical and Spatial
Properties of CRN
3.1. e Small-World Property of CRN. e small-world
property is a ubiquitous characteristic of a complex network,
Journal of Advanced Transportation
1
23
4
5
6
7
8
9
Route 1
Route 2
Station
(a)
1
2
3
4
5
6
7
8
9
Route 1
Route 2
Station
(b)
F : (a) A railway network consists of two train routes, train route  in orange and train route  in blue. (b) e P-space network formed
by the two train routes in Figure (a), where the orange lines denote node pairs connected through the train route , and the blue lines denote
node pairs connected through train route .
as shown in other complex systems. A small-world network
is a network with a short average path length and a large
clustering coecient. Small average path length exists in
random graphs, and a large clustering coecient can be found
in regular lattices but not in random graphs. e small-world
property measures the transportation eciency of a network
at the global scale.
e average path length of CRN is ., which means
passengers only need to take three trains on average to travel
between any pairs of the  cities of China. e maximum
shortest path length between a pair of cities in CRN is , and
city pairs of this type are rare. Inaddition, % of the city pairs
are connected by two or fewer steps (topological distance),
conrming that CRN is a mature and ecient transportation
infrastructure. In China, railroads and airlines are in erce
competition. e average path length of CRN is similar to
that (.) of the Chinese airline transportation system [].
However, CRN covers many more cities ( cites) than the
Chinese airline transportation system ( cities), oering
valuableservicetoremoteandsmallcities.
As previously noted, the clustering coecient can be used
to describe the cliquishness of CRN. e clustering coecient
of CRN is ., which is substantially larger than a random
network (𝑟𝑎𝑛𝑑𝑜𝑚 ≈ 0.095) of the same size (the same number
of nodes and edges). We can conclude from these ndings
that CRN is a small-world network.
3.2. e Scale-Free Structure of CRN. Ascale-freenetwork
is one with a power-law degree distribution of p() ∝
c−𝑟 with an exponent parameter .Suchanetworkis
regarded as robust to random node failure because a large
portion of nodes have few connections with others. However,
information of the edge-weight is crucial for analyzing CRN
as a weighted complex network. We thus analyze the scale-
free property of CRN via its edge-weight information. e
edge-weights of CRN were counted using train ow infor-
mation (the number of trains between city pairs), resulting
in an average value of  and a range of  to . Around
% of edges have larger weights than the average weight.
is phenomenon is referred to as the “/” rule or the
CRN
Power
f(t)=1.03t-1.07
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(≥ t)
100 200 300 400 500 6000
Train ow, t
F : e cumulative distribution of train ow (edge-weight) in
CRN.
Pareto principle in other transportation systems []. e
statistical distribution of the edge-weight was tted to reveal
the pattern of train ow and was shown in Figure . e
obtained cumulative distribution exhibits a long-tail with few
extreme values, indicating a scale-free structure. A nonlinear
least square method was applied to the scaling parameter
estimator, and the results conrmed that the cumulative
distribution obeys a power-law function with the exponent
parameter  = 1.07.
e power-law distribution shows that a large portion of
edges reect low train ow intensity, implying a high level
of heterogeneity in city connections within CRN. Similar
heterogeneity of trac ows was also found in other trans-
portation systems, such as the Singapore railway system []
and the bus transportation network in China []. Generally,
train ows are appropriate indicators of passenger ows
Journal of Advanced Transportation
70 80 90 100 110 120 130 140
20
25
30
35
40
45
50
55
10 20 60 200 600
Cities
5
4
3
2
1
67
8
9
13
14
15
16
26
10
11
12
18
17
21
22
23
24
25
19
20
27
28
29
30
Longitude
Latitude
F : e spatial distribution of train ows in CRN. e size
of the nodes reects the degree of the city, and the colors of the
lines indicate train ows. e black dashed line denotes the Hu
Line, which demarcates the concentration of population of China.
Cities along the four vertical and four horizontal railway corridors
are marked as ABeijing, BTianjin, CQinhuangdao, DShenyang,
EHa’erbin, FTa i y u an, GShijiazhuang, HJinan, 0Qingdao,
1Urumch i, 2Lanzhou, 3Xi’an, 4Zhengzhou, 5Xuzhou, 6
Nanjing, 7Shanghai, 8Hangzhou, 9Wu h an, :Chongqing, ;
Chengdu, Nanchang, œChangsha, Huaihua, XGuiyang,
Kunming, YNingbo, ZFuzhou, Xiamen, Shenzhen,
Guangzhou.
between node pairs [], and thus, we may infer that passenger
ows between city pairs also exhibit signicant heterogeneity
in CRN.
3.3. Spatial Heterogeneity and Hierarchy of CRN. Railway
networks are spatial networks embedded in the geographical
space. e above statistical analyses reveal the heterogeneity
of CRN but fail to illustrate the spatial characteristics of
the network. To uncover the underlying spatial structure,
we map CRN into a connected graph in a geographic
coordinate system (see Figure ). CRN shows a clear dier-
ence between the southeast and northwest sides of the Hu
Line, a demarcation line for China’s population proposed by
the prominent geographer Hu Huanyong. e southeastern
terrain of China, dominated by plains of low elevation, has a
high population density and intense economic activity, while
the northwestern part of China is dominated by plateaus
and mountains, resulting in a low population density and an
underdeveloped economy. e unbalanced populations and
economic development help explain the uneven distribution
of CRN in southeast and northwest China.
To further explore the spatial heterogeneity and hierarchy
of CRN, we constructed  subnetworks (a subnetwork
includes cities in a province and the connections between
the cites) by province in China. e inner network density
(density= M/(N(N-)), M is the number of edges between
cities in the province and N is the number of cities in
the province) of each subnetwork, namely, the connection
strength between cities inside a provincial administrative
T : Densities of subnetworks divided by provinces.
Province Region Density
Liaoning .
Jilin Northeast China .
Heilongjiang .
Inner Mongolia .
Shanxi North China .
Hebei .
Ningxia .
Shaanxi .
Gansu NorthwestChina .
Sinkiang .
Qinghai .
Hainan .
Guangxi South China .
Guangdong .
Hubei .
Hunan Central C hina .
Henan .
Fujian
East China
.
Zhejiang .
Shandong .
Anhui .
Jiangxi .
Jiangsu .
Tibet
SouthwestChina
.
Guizhou .
Sichuan .
Yunnan .
division, was calculated to measure the spatial heterogeneity
of CRN (see Table ). e densities of the subnetworks in
southwest and northwest China, including Sichuan, Guizot,
Yunnan, Tibet, Gansu, Sinkiang, and Qinghai provinces, are
., ., ., ., ., ., and ., respec-
tively, substantially smaller than those of the provinces of east
China. is conrms the observations in Figure . Moreover,
we applied the Gini coecient (G-value) to measure the
spatial heterogeneity of CRN [], which is formulated as
follows:
G=+1
−12
(−1
)
𝑛
𝑖=1 𝑖𝑖,()
where denotes the number of edges in the network; 𝑖is
the edge-weight, namely, train ow; 𝑖represents the rank
of edge sorted by edge-weight in descending order; and is
the mean value of edge-weight. G-value ranges from  to ,
and a larger G-value indicates a more heterogeneous network.
e G-value of CRN was calculated as ., indicating the
signicant heterogeneity of intercity train ows in CRN.
e scale-free property of train ows also implies a
hierarchical structure for CRN. Figure  shows that a small
number of city pairs are intensely connected by hundreds
of trains (warm color lines), and the majority of city pairs
Journal of Advanced Transportation
are connected by limited numbers of trains (cool color lines
and gray lines). Notably, cities with good connectivity and
intensive trac ows form the framework of CRN and
occur along “the four vertical and four horizontal” railway
corridors. e four vertical lines include the Beijing-Shijiazhu
ang-Zhengzhou-Wuhan-Changsha-Guangzhou line, the Bei-
jing-Tianjin-Jinan-Hefei-Nanjing-Shanghai line, the Beijing-
Qinhuangdao-Shenyang-Ha’erbin line, and the Shanghai-
Hangzhou-Fuzhou-Xiamen-Guangzhou line. e four hori-
zontal lines are the Xuzhou-Shangqiu-Zhengzhou-Xian-Ba-
oji-Lanzhou, the Qingdao-Jinan-Taiyuan-Shijiazhuang line,
the Shanghai-Nanjing-Wuhan-Chongqing-Chengdu line,
and the Shanghai-Hangzhou-Changsha-Guiyang-Kunming
line. e interactions of the four vertical and four horizontal
lines are the core cities of CRN with the most connections
and train trac, such as Beijing, Shanghai, Zhengzhou,
Wuhan, and Changsha. All these cities are national or
regional economic and political centers, indicating that
strong political factors inuence the hierarchical structure of
CRN.
4. The Nodal Centralities of CRN
4.1. Degree and Strength. To gain deeper insights into the
structureandevolutionofCRN,wecalculatedthedegree
centrality (connectivity) and strength centrality of each city.
For CRN, nodal degree ranges from  to , with an average
value of . % of the cities are less connected than average.
e average value of nodal strength was  with a range from
 to . % of the cities have lower strength than average
train trac. Figures (a) and (b) display the cumulative
distributions of nodal degree and strength, both of which
approximately follow an exponential decaying. is suggests
that CRN evolves randomly, unlike airline networks, whose
degree distributions tend to follow a power-law distribution
[,,].
A possible explanation for the dierence in degree and
strength distributions between CRN and airline networks
stems from the organizations of the two types of net-
works. Generally, airline transportation networks adopt a
hub-and-spoke service strategy, and expansions of airline
networks coincide with the preferential attachment model,
which draws expansions to more connected hubs; this is
known as the rich-club eect []. Moreover, two airports are
usually connected by a nonstop air-route, and intermediate
airports are rare. However, to ensure service to more cities, a
train route usually covers – railway stations. In P-space
representation, one route will generate a fully connected
graph, which raises the degree and strength of intermediate
stations. In addition, a railway station can only handle a
limited number of railway tracks and trains, resulting in
relatively homogeneous distributions instead of power-law
distributions.
4.2. Betweenness and Closeness. Special attention should also
be paid to nodal betweenness and closeness centralities,
which measure the global centrality and accessibility of nodes
in the network, respectively. Nodal betweenness ranges from
 to . in CRN, with an average value of .. % of the
nodes exhibit lower value than average betweenness, which
suggests that few powerful nodes have absolute control power
over the whole network. % of the cities have a betweenness
value of , meaning that there is no shortest path passing
through them. All of these nodes are terminal cities of railway
routes, and most of the cities are located in less-developed
regions with low population densities. Cities with high
betweenness (e.g., the top ) are mainly provincial capitals
with advanced economies and high population densities.
ese cities compose the core layer of CRN and account
for most of the transfers in the network. e cumulative
distribution of nodal betweenness is plotted in Figure (a),
which reveals a monotonically decreasing trend that can be
approximated by an exponential function.
e closeness of nodes ranges from . to ., with an
average value of .. Figure (b) presents the cumulative
distribution of nodes’ closeness, which shows an unfamiliar
pattern of an inverse “S” curve. is shows that few cities
have extremely weak closeness (only % of the cities have
closeness under .). is nding further conrms that CRN
is an ecient infrastructure network.
4.3. e Relationships between Degree and the Other ree
Measures. e top  cities ranked by degree are listed
in Table . Notably, the rankings of cities vary with the
measure used. For example, Shanghai ranks second in
degree, strength, and closeness but twelh in betweenness.
We exploited relationships between degree and strength,
betweenness, and closeness to uncover further information
on the topological structure of CRN. Figure (a) shows that
the relationship between nodal degree and strength can be
tted by a power-law function, but not by the expected linear
one. A similar trait was also found in the US intercity airline
transportation network []. e main reason behind this
is that a small number of hubs have many connections and
handle much more trac ow than the peripheral nodes.
Nodal degree is positively correlated with betweenness
and closeness (see Figures (b) and (c)), which can be tted
by a power function and a linear function (for degree ),
respectively. Notably, variations in degree and betweenness
are less consistent. An important question explored by
previous studies on the relationship between nodal degree
and betweenness is whether the most connected nodes are
also the most central. In many complex networks, including
randomized networks [], Internet networks, and social
networks [], nodal degree and betweenness have a strong
linear relation. In contrast, CRN shows some anomalies: cer-
tain cities have small degree and large betweenness (circled
by red ellipses in Figure (b)).
To explore the underlying reasons for anomalies, cities are
classied into ve tiers based on nodal degree and between-
ness, respectively (see Figures (a) and (b)). Prominent
dierences can be identied in the two maps. ese dier-
ences identied between the two classications indicate the
strong inuences of socioeconomy, politics, and geography
in the development of CRN. For example, Lasa and Haikou
are in the bottom-tiers in terms of the nodal degree but
belong to the top-tier in terms of nodal betweenness. Lasa
and Haikou are located in the remote and peripheral area of
Journal of Advanced Transportation
T : e top  cities ranked by degree.
City Degree Rank Streng th Rank Betweenness Rank Closeness Rank
Beijing     . . 
Shanghai    .  .
Zhengzhou   . .
Xian     . .
Hangzhou     .  .
Tianjin     .  .
Wuhan   .  .
Nanchang     .  .
Xuzhou    .  . 
Guangzhou    . .
Nanjing    .  . 
Shenyang    . . 
Shijiazhuang     .  . 
Jinan     .  . 
Jinzhou     . . 
Bengbu     .  . 
Qinhuangdao     .  . 
Suzhou     .  . 
Shangqiu     .  . 
Zhuzhou     .  . 
CRN
Exponential
f(d)=0.97exp(-0.008d)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(≥ d)
100 200 300 400 500 6000
Degree, d
(a)
f(s)=0.87exp(-0.002s)
1000 2000 3000 4000 5000 60000
Strength, s
CRN
Exponential
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(≥ s)
(b)
F : (a) e cumulative distribution of the nodal degree in CRN. (b) e cumulative distribution of the nodal strength in CRN.
China and are weakly connected with other cities. However,
those two cities are economic hubs, local political centers,
and respective gateways to Tibet Autonomous Region and
Hainan Province. us, they play critical roles in connecting
small cities scattered around them to other cities in the
network.
5. The Role of Cities in CRN
Centrality measures, including degree, strength, between-
ness, and closeness, reect dierent aspects of cities’ impor-
tance in CRN. As noted above, the weak connection of a
city does not imply unimportance, because the city may have
high betweenness and play a bridging role. A city exhibiting
good connectivity is not necessarily globally central in the
network. at is, a single measure fails to capture the overall
importance of any city in CRN. A typical example can be
obtained through a comparison of Kunming and Jinzhou.
e former is the economic center and capital city of Yunnan
Province as well as the gateway to southwest China, whereas
the latter is a less-developed, medium-sized noncapital city in
central China. Kunming has the second-highest betweenness
centrality (.), but its connectivity () is weaker than
that of Jinzhou (), whose betweenness is only ..
Journal of Advanced Transportation
f(b)=0.7exp(-1742b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(≥ b)
0.01 0.02 0.03 0.04 0.050
Betweenness, b
CRN
Exponential
(a)
CRN
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(≥ c)
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70.3
Closeness, c
(b)
F : (a) e cumulative distribution of the nodal betweenness in CRN. (b) e cumulative distribution of the nodal closeness in CRN.
Overall, Kunming plays a more important role in CRN than
Jinzhou. However, this cannot be easily discerned using only
commonly used degree. Luan et al. [] proposed a multiple-
criteria indicator based on the degree, betweenness, and
closeness centralities to capture nodes’ importance and the
hierarchical structure of a network. Inspired by this idea, we
propose an integrated measure based on the four centrality
measures and dene it as a hub indicator to quantify the
globalrolesofcitiesinthenetwork.ismeasureisasfollows:
𝑖=𝐷()−
𝐷,𝑚𝑖𝑛
𝐷,𝑚𝑎𝑥 −
𝐷,𝑚𝑖𝑛 +𝑆()−
𝑆,𝑚𝑖𝑛
𝑆,𝑚𝑎𝑥 −
𝑆,𝑚𝑖𝑛
+𝐵()−
𝐵,𝑚𝑖𝑛
𝐵,𝑚𝑎𝑥 −
𝐵,𝑚𝑖𝑛 +𝐶()−
𝐶,𝑚𝑖𝑛
𝐶,𝑚𝑎𝑥 −
𝐶,𝑚𝑖𝑛
()
where ,,,andare respective weights of the unied
nodal degree, strength, betweenness, and closeness. e
values of ,,,andare calculated using the coecient
of variation method, a data-driven method for measuring
quantity weights (see the references for the details on the
procedures for the coecient of variation method) [, ].
,,,andare calculated to be ., ., ., and .,
respectively, and 𝑖is in the range of . to . with an
average value of ..
Following the classication rules suggested by Guimer´
a
et al. [] and Du et al. [], we divided the cities into four
categories based on the value of 𝑖via the k-means clustering
algorithm: (1) national core cities with 0.62 ≤ 𝑖<0.98,
(2) bridge cities with 0.42 ≤ 𝑖<0.62,(3) peripheral
cities with 0.26 ≤ 𝑖<0.42,and(4) ultraperipheral cities
with 0≤
𝑖<0.26. e spatial distribution of the city
categorizations is plotted in Figure . Notably, core and bridge
cities are mainly national or local economic and political
centers scattered along the “four vertical and four horizontal”
railway corridors. Most of the peripheral and ultraperipheral
cities are located in remote or peripheral regions and are less-
developed. is categorization is signicant and consistent
with the organization and evolution of CRN.
ere are  cities and  edges (.% of the total
edges) in the core layer,  cities and , edges (.%)
in the bridge layer,  cities and , edges (.%) in
the peripheral layer, and  cities and , edges (.%)
in the ultraperipheral layer. Moreover, there are , edges
(.%) between the core layer and the bridge layer; ,
edges (.%) between the core layer and the peripheral
or ultraperipheral layer; , edges (.%) between the
bridge layer and the peripheral or ultraperipheral layer; and
, edges (.%) between the peripheral layer and the
ultraperipheral layer. A remarkable nding from this analysis
is that stable connections hold between dierent layers in
CRN, which is substantially dierent from the Chinese airline
network, where most connections (%) are within the
core layer and minimal connections (.%) exist between
thecorelayerandtheperipherallayer[].isnding
further demonstrates that CRN is a mature and ecient
infrastructure network.
6. Conclusion
We investigated CRN by modeling it as a complex weighted
network. Our ndings suggest that CRN is a small-world,
scale-free infrastructure network with a small average path
length (.) and a large cluster coecient (.). Unlike
other complex networks such as the Internet and social and
biological networks, the distributions of nodal centralities of
Journal of Advanced Transportation
CRN
Power
f(d)=0.04d1.85
0
1000
2000
3000
4000
5000
6000
7000
Strength, s
100 200 300 400 500 600 7000
Degree, d
(a)
f(d)=1.04d3.38
0
0.01
0.02
0.03
0.04
0.05
Betweenness, b
100 200 300 400 500 600 7000
Degree, d
CRN
Power
(b)
CRN
Linear
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Closeness, c
100 200 300 400 500 600 7000
Degree, d
(c)
F : (a) e relationship between nodal degree and strength in CRN. (b) e relationship between nodal degree and betweenness in
CRN. (c) e relationship between nodal degree and closeness in CRN.
CRN, including degree, strength, betweenness, and closeness,
exhibit patterns of exponential functions or an inverted
“S” shape. Nodal degree is positively correlated with nodal
strength, betweenness, and closeness. However, our analysis
reveals that the most connected cities are not necessarily the
most central because of the inuences of social, political, and
geographical factors.
Train trac in CRN follows a power-law distribution,
implying heterogeneity and hierarchy of the network. To
illustrate the underlying reasons for this pattern, we mapped
a topological connectivity graph of CRN in a geographic
coordinate system. Our ndings show that uneven popula-
tion distributions and economic clout account for the uneven
distribution of CRN services between southeast and north-
west China. e “four vertical and four horizontal” railway
corridors establish major connections and train ows of CRN
and pass through cities that are national or local political
centers with advanced economies and dense populations.
is indicates that the uneven distribution of CRN services
reects a strong political inuence.
Journal of Advanced Transportation
top 15% -30% of cities
top 30% -50% of cities
top 50% -100% of cities
top 5% of cities
top 5% -15% of cities
Lasa
Haikou
20
25
30
35
40
45
50
55
Latitude
80 90 100 110 120 130 14070
Longitude
(a)
top 5% of cities
top 15%-30% of cities
top 30%-50% of cities
top 50%-100% of cities
top 5%-15% of cities
Lasa
Haikou
20
25
30
35
40
45
50
55
Latitude
80 90 100 110 120 130 14070
Longitude
(b)
F : (a) e spatial distribution of cities classied by nodal degree value in descending order. (b) e spatial distribution of the cities
classied by nodal betweenness value in descending order.
core cities
bridge cities
peripheral cities
ultraperipheral cities
20
25
30
35
40
45
50
55
Latitude
80 90 100 110 120 130 14070
Longitude
F : e spatial distribution of the cities in dierent categories
according to the proposed integrated index.
Nodal degree, strength, betweenness, and closeness quan-
tify the importance of cities in CRN from dierent perspec-
tives. However, no single measure can uncover the role of
cities on a global scale. us, we proposed an integrated indi-
cator that reveals the multilayer structure of CRN. It classied
cities into four categories (core cities, bridge cities, peripheral
cities, and ultraperipheral cities). Unlike the Chinese airline
network, CRN has remarkably stable connections between
dierent layers of the network, demonstrating the CRN’s
accessibility and eciency.
Our research has some limitations. China’s railway trans-
portation system comprises dierent types of trains, such as
G-number, D-number, C-number, Z-number, T-number, and
K-number, with varying speeds and capacities. is research
focuses on the connectivity of CRN and represents trains
between cities as weighted edges in the topological network
without considering train type. Assigning dierent weights
to dierent trains will enable a more comprehensive analysis
but requires substantial eorts. is is now included in our
agenda for future research. Another direction is to examine
networks formed by dierent types of trains separately, for
example, China’s high-speed railway network, comprised of
G-number and D-number trains. Preliminary results suggest
that China’s high-speed railway network exhibits certain
properties distinct from those of CRN as a whole. Further
analyses in this direction are in progress.
Data Availability
e data used to support the ndings of this study are
available from the corresponding author upon request.
Disclosure
e authors are ordered alphabetically.
Conflicts of Interest
e authors declare that they have no conicts of interest.
 Journal of Advanced Transportation
Acknowledgments
is work was supported by the National Natural Science
Foundation of China (Grants No. , , and
).
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