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: Roc´ıo 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 eciency 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 signicant 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 conrm that stable connections hold between dierent 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 system’s small-world character. Soh et al. [] showed that

Singapore’s railway network is almost fully connected, and its

hub nodes experience disproportionately heavy trac. e

Chinese railway network (CRN) is one of the largest railway

networks in the world and signicantly contributes to the

country’s development []. However, the structural perfor-

mance of CRN has not been given sucient 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

dened 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 Coecient. e clustering coecient [] of

node is as follows:

𝑖=2𝑖

𝑖𝑖−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 dened

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 dened as if there is a connection between

nodes and ,andotherwise.

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 coecient. Small average path length exists in

random graphs, and a large clustering coecient can be found

in regular lattices but not in random graphs. e small-world

property measures the transportation eciency 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),

conrming that CRN is a mature and ecient 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), oering

valuableservicetoremoteandsmallcities.

As previously noted, the clustering coecient can be used

to describe the cliquishness of CRN. e clustering coecient

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)=1.03∗t-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 conrmed 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 reect low train ow intensity, implying a high level

of heterogeneity in city connections within CRN. Similar

heterogeneity of trac 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 reects 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 signicant 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 dier-

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 conrms the observations in Figure . Moreover,

we applied the Gini coecient (G-value) to measure the

spatial heterogeneity of CRN [], which is formulated as

follows:

G=+1

−1−2

(−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

signicant 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 trac 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 trac, such as Beijing, Shanghai, Zhengzhou,

Wuhan, and Changsha. All these cities are national or

regional economic and political centers, indicating that

strong political factors inuence 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 trac. 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 dierence 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 eect []. 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 conrms that CRN

is an ecient 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 twelh 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 trac 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

classied into ve tiers based on nodal degree and between-

ness, respectively (see Figures (a) and (b)). Prominent

dierences can be identied in the two maps. ese dier-

ences identied between the two classications indicate the

strong inuences 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)=0.97∗exp(-0.008∗d)

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)=0.87∗exp(-0.002∗s)

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, reect dierent 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)=0.7∗exp(-1742∗b)

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 dene 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 unied

nodal degree, strength, betweenness, and closeness. e

values of ,,,andare calculated using the coecient

of variation method, a data-driven method for measuring

quantity weights (see the references for the details on the

procedures for the coecient of variation method) [, ].

,,,andare calculated to be ., ., ., and .,

respectively, and 𝑖is in the range of . to . with an

average value of ..

Following the classication 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 signicant 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 dierent layers in

CRN, which is substantially dierent from the Chinese airline

network, where most connections (%) are within the

core layer and minimal connections (.%) exist between

thecorelayerandtheperipherallayer[].isnding

further demonstrates that CRN is a mature and ecient

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 coecient (.). 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)=0.04∗d1.85

0

1000

2000

3000

4000

5000

6000

7000

Strength, s

100 200 300 400 500 600 7000

Degree, d

(a)

f(d)=1.04∗d3.38

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 inuences of social, political, and

geographical factors.

Train trac 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

reects a strong political inuence.

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 classied by nodal degree value in descending order. (b) e spatial distribution of the cities

classied 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 dierent categories

according to the proposed integrated index.

Nodal degree, strength, betweenness, and closeness quan-

tify the importance of cities in CRN from dierent 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 classied

cities into four categories (core cities, bridge cities, peripheral

cities, and ultraperipheral cities). Unlike the Chinese airline

network, CRN has remarkably stable connections between

dierent layers of the network, demonstrating the CRN’s

accessibility and eciency.

Our research has some limitations. China’s railway trans-

portation system comprises dierent 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 dierent weights

to dierent trains will enable a more comprehensive analysis

but requires substantial eorts. is is now included in our

agenda for future research. Another direction is to examine

networks formed by dierent 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 conicts 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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