Journal of Advanced Transportation
Volume 2019 (2019), Article ID 3928260, 10 pages
https://doi.org/10.1155/2019/3928260

Research Article

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

Weiwei Cao,¹ Xiangnan Feng,¹ Jianmin Jia,² and Hong Zhang³

¹School of Economic and Management, Southwest Jiaotong University, Chengdu 610031, China
²School of Economic and Management, The 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; fengxiangnan123@gmail.com

Received 23 June 2018; Revised 18 November 2018; Accepted 4 December 2018; Published 3 February 2019

Academic Editor: Rocío de Oña

Copyright © 2019 Weiwei Cao et al. This 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.

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.

1. Introduction

Transportation networks have enormous impacts on national economic and social activities [1, 2]. As argued by Alderighi et al. [3], the structure of a network shapes operational strategy and service quality of the system. Thus, 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 [4–6], shipping networks [7], subway networks [8], bus networks [9, 10], and railway networks [11–13].

Like other transportation systems, railway networks are closely linked to sustainable regional development [14]. Particular attention has been paid to the topological properties of railway networks. Sen et al. [12] first employed complex network theory to study India’s railway network, revealing the system’s small-world character. Soh et al. [13] showed that Singapore’s railway network is almost fully connected, and its hub nodes experience disproportionately heavy traffic. The Chinese railway network (CRN) is one of the largest railway networks in the world and significantly contributes to the country’s development [15]. However, the structural performance of CRN has not been given sufficient attention in authoritative journals; previous research has focused mainly on the structural properties of CRN through pure statistical analyses of stations’ connections [11, 16–18] 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 78,000 km in 2007 to 127,000 km in 2017. Thus, the structure of CRN may have changed substantially and thus needs a thorough reexamination. 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 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. This measure provides meaningful insights into the national roles of cities in CRN, which helps reveal the multilayer structure of CRN.

The rest of this paper is structured as follows: Section 2 introduces data and structure measures. Section 3 reports the statistical and spatial properties of CRN at a global scale. Section 4 presents the centrality measures of nodes and explores their relationships. Section 5 proposes an integrated measure to reveal the role of each node in CRN. Section 6 concludes the paper.

2. Data Sources and Network Structure Measures

2.1. Data Sources

The data set analyzed was provided by a railway bureau of China. It includes information on all railway stations and over 7,000 domestic scheduled passenger trains in China in 2017. For transportation networks, a widely used methodology for abstracting a system into a complex network is the P-space method [19], illustrated in Figure 1. The 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 [4], 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 nodes (i.e., cities) and 67,594 edges.

(a)

(b)

Figure 1: (a) A railway network consists of two train routes, train route 1 in orange and train route 2 in blue. (b) The P-space network formed by the two train routes in Figure 1(a), where the orange lines denote node pairs connected through the train route 1, and the blue lines denote node pairs connected through train route 2.

2.2. Network Structure Measures

We applied a variety of topological measures to explore the structural characteristics of CRN. The first two measures summarized the global scale properties of CRN, and the following four measures characterized nodes’ centralities in the network.

2.2.1. Average Path Length

Average path length [20] is defined as the average value of the shortest path lengths between all node pairs in a network:

where is the number of nodes and is the shortest path length between node and node .

2.2.2. Clustering Coefficient

The clustering coefficient [20] of node is as follows:

where is the number of neighbors connected to node , is the actual number of edges connecting the neighbors, and ( − 1)/2 is the largest possible number of edges between these neighbors.

2.2.3. Degree Centrality

The degree of a node [21] is defined as the number of neighbors in the network connected to that node. It is represented as follows:

where represents the set of all nodes in the network except node , and is defined as 1 if there is a connection between nodes and , and 0 otherwise.

2.2.4. Strength Centrality

As an extension of degree, strength centrality combines the connectivity and train flow information of a node [21]. Strength is formalized as follows:

where represents the weight of the edge between nodes and .

2.2.5. Betweenness Centrality

Betweenness centrality [21] is represented as follows.

Here, counts the number of possible shortest paths between nodes and , denotes the number of shortest paths between nodes and that pass through node , and / represents the proportion of the shortest paths between nodes and that pass through node .

2.2.6. Closeness Centrality

The closeness centrality [21] of node is represented as follows:

where is the number of nodes in the network, represents the set of all nodes in the network except node , and is the shortest path length between nodes and .

3. The Statistical and Spatial Properties of CRN

3.1. The Small-World Property of CRN

The small-world property is a ubiquitous characteristic of a complex network, as shown in other complex systems. A small-world network is a network with a short average path length and a large clustering coefficient. Small average path length exists in random graphs, and a large clustering coefficient can be found in regular lattices but not in random graphs. The small-world property measures the transportation efficiency of a network at the global scale.

The average path length of CRN is 2.21, which means passengers only need to take three trains on average to travel between any pairs of the 1192 cities of China. The maximum shortest path length between a pair of cities in CRN is 5, and city pairs of this type are rare. In addition, 71% of the city pairs are connected by two or fewer steps (topological distance), confirming that CRN is a mature and efficient transportation infrastructure. In China, railroads and airlines are in fierce competition. The average path length of CRN is similar to that (2.23) of the Chinese airline transportation system [22]. However, CRN covers many more cities (1192 cites) than the Chinese airline transportation system (203 cities), offering valuable service to remote and small cities.

As previously noted, the clustering coefficient can be used to describe the cliquishness of CRN. The clustering coefficient of CRN is 0.68, which is substantially larger than a random network () of the same size (the same number of nodes and edges). We can conclude from these findings that CRN is a small-world network.

3.2. The Scale-Free Structure of CRN

A scale-free network is one with a power-law degree distribution of with an exponent parameter . Such a network is 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. The edge-weights of CRN were counted using train flow information (the number of trains between city pairs), resulting in an average value of 5 and a range of 1 to 623. Around 20% of edges have larger weights than the average weight. This phenomenon is referred to as the “20/80” rule or the Pareto principle in other transportation systems [22]. The statistical distribution of the edge-weight was fitted to reveal the pattern of train flow and was shown in Figure 2. The 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 confirmed that the cumulative distribution obeys a power-law function with the exponent parameter .

Figure 2: The cumulative distribution of train flow (edge-weight) in CRN.

The power-law distribution shows that a large portion of edges reflect low train flow intensity, implying a high level of heterogeneity in city connections within CRN. Similar heterogeneity of traffic flows was also found in other transportation systems, such as the Singapore railway system [13] and the bus transportation network in China [10]. Generally, train flows are appropriate indicators of passenger flows between node pairs [8], and thus, we may infer that passenger flows between city pairs also exhibit significant heterogeneity in CRN.

3.3. Spatial Heterogeneity and Hierarchy of CRN

Railway networks are spatial networks embedded in the geographical space. The 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 3). CRN shows a clear difference between the southeast and northwest sides of the Hu Line, a demarcation line for China’s population proposed by the prominent geographer Hu Huanyong. The 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. The unbalanced populations and economic development help explain the uneven distribution of CRN in southeast and northwest China.

Figure 3: The spatial distribution of train flows in CRN. The size of the nodes reflects the degree of the city, and the colors of the lines indicate train flows. The 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 ① Beijing, ② Tianjin, ③ Qinhuangdao, ④ Shenyang, ⑤ Ha’erbin, ⑥ Taiyuan, ⑦ Shijiazhuang, ⑧ Jinan, ⑨ Qingdao, ⑩ Urumchi, ⑪ Lanzhou, ⑫ Xi’an, ⑬ Zhengzhou, ⑭ Xuzhou, ⑮ Nanjing, ⑯ Shanghai, ⑰ Hangzhou, ⑱ Wuhan, ⑲ Chongqing, ⑳ Chengdu, ㉑ Nanchang, ㉒ Changsha, ㉓ Huaihua, ㉔ Guiyang, ㉕ Kunming, ㉖ Ningbo, ㉗ Fuzhou, ㉘ Xiamen, ㉙ Shenzhen, ㉚ Guangzhou.

To further explore the spatial heterogeneity and hierarchy of CRN, we constructed 28 subnetworks (a subnetwork includes cities in a province and the connections between the cites) by province in China. The inner network density (density= 2M/(N(N-1)), 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 division, was calculated to measure the spatial heterogeneity of CRN (see Table 1). The densities of the subnetworks in southwest and northwest China, including Sichuan, Guizot, Yunnan, Tibet, Gansu, Sinkiang, and Qinghai provinces, are 0.038, 0.061, 0.025, 0.006, 0.064, 0.069, and 0.038, respectively, substantially smaller than those of the provinces of east China. This confirms the observations in Figure 3. Moreover, we applied the Gini coefficient (G-value) to measure the spatial heterogeneity of CRN [23], which is formulated as follows:

where denotes the number of edges in the network; is the edge-weight, namely, train flow; represents the rank of edge sorted by edge-weight in descending order; and is the mean value of edge-weight. G-value ranges from 0 to 1, and a larger G-value indicates a more heterogeneous network. The G-value of CRN was calculated as 0.77, indicating the significant heterogeneity of intercity train flows in CRN.

Table 1: Densities of subnetworks divided by provinces.

Province
Region
Density

Liaoning

0.554

Jilin
Northeast China
0.27

Heilongjiang

0.203

Inner Mongolia

0.147

Shanxi
North China
0.106

Hebei

0.165

Ningxia

0.373

Shaanxi

0.129

Gansu
Northwest China
0.064

Sinkiang

0.069

Qinghai

0.038

Hainan

0.532

Guangxi
South China
0.13

Guangdong

0.158

Hubei

0.151

Hunan
Central China
0.11

Henan

0.106

Fujian
East China
0.234

Zhejiang
0.221

Shandong
0.184

Anhui
0.162

Jiangxi
0.152

Jiangsu
0.147

Tibet
Southwest China
0.006

Guizhou
0.061

Sichuan
0.038

Yunnan
0.025

The scale-free property of train flows also implies a hierarchical structure for CRN. Figure 3 shows that a small number of city pairs are intensely connected by hundreds of trains (warm color lines), and the majority of city pairs are connected by limited numbers of trains (cool color lines and gray lines). Notably, cities with good connectivity and intensive traffic flows form the framework of CRN and occur along “the four vertical and four horizontal” railway corridors. The four vertical lines include the Beijing-Shijiazhuang-Zhengzhou-Wuhan-Changsha-Guangzhou line, the Beijing-Tianjin-Jinan-Hefei-Nanjing-Shanghai line, the Beijing-Qinhuangdao-Shenyang-Ha’erbin line, and the Shanghai-Hangzhou-Fuzhou-Xiamen-Guangzhou line. The four horizontal lines are the Xuzhou-Shangqiu-Zhengzhou-Xian-Baoji-Lanzhou, the Qingdao-Jinan-Taiyuan-Shijiazhuang line, the Shanghai-Nanjing-Wuhan-Chongqing-Chengdu line, and the Shanghai-Hangzhou-Changsha-Guiyang-Kunming line. The interactions of the four vertical and four horizontal lines are the core cities of CRN with the most connections and train traffic, such as Beijing, Shanghai, Zhengzhou, Wuhan, and Changsha. All these cities are national or regional economic and political centers, indicating that strong political factors influence the hierarchical structure of CRN.

4. The Nodal Centralities of CRN

4.1. Degree and Strength

To gain deeper insights into the structure and evolution of CRN, we calculated the degree centrality (connectivity) and strength centrality of each city. For CRN, nodal degree ranges from 1 to 673, with an average value of 113. 64% of the cities are less connected than average. The average value of nodal strength was 504 with a range from 2 to 6694. 73% of the cities have lower strength than average train traffic. Figures 4(a) and 4(b) display the cumulative distributions of nodal degree and strength, both of which approximately follow an exponential decaying. This suggests that CRN evolves randomly, unlike airline networks, whose degree distributions tend to follow a power-law distribution [4, 6, 24].

(a)

(b)

Figure 4: (a) The cumulative distribution of the nodal degree in CRN. (b) The cumulative distribution of the nodal strength in CRN.

A possible explanation for the difference in degree and strength distributions between CRN and airline networks stems from the organizations of the two types of networks. 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 effect [3]. 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 10–20 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 0 to 0.053 in CRN, with an average value of 0.001. 82% of the nodes exhibit lower value than average betweenness, which suggests that few powerful nodes have absolute control power over the whole network. 13% of the cities have a betweenness value of 0, 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 30) are mainly provincial capitals with advanced economies and high population densities. These cities compose the core layer of CRN and account for most of the transfers in the network. The cumulative distribution of nodal betweenness is plotted in Figure 5(a), which reveals a monotonically decreasing trend that can be approximated by an exponential function.

(a)

(b)

Figure 5: (a) The cumulative distribution of the nodal betweenness in CRN. (b) The cumulative distribution of the nodal closeness in CRN.

The closeness of nodes ranges from 0.29 to 0.69, with an average value of 0.46. Figure 5(b) presents the cumulative distribution of nodes’ closeness, which shows an unfamiliar pattern of an inverse “S” curve. This shows that few cities have extremely weak closeness (only 20% of the cities have closeness under 0.4). This finding further confirms that CRN is an efficient infrastructure network.

4.3. The Relationships between Degree and the Other Three Measures

The top 20 cities ranked by degree are listed in Table 2. Notably, the rankings of cities vary with the measure used. For example, Shanghai ranks second in degree, strength, and closeness but twelfth in betweenness. We exploited relationships between degree and strength, betweenness, and closeness to uncover further information on the topological structure of CRN. Figure 6(a) shows that the relationship between nodal degree and strength can be fitted 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 [24]. The main reason behind this is that a small number of hubs have many connections and handle much more traffic flow than the peripheral nodes.

Table 2: The top 20 cities ranked by degree.

City
Degree
Rank
Strength
Rank
Betweenness
Rank
Closeness
Rank

Beijing
673
1
5749
4
0.054
1
0.696
1

Shanghai
549
2
6694
1
0.014
12
0.645
2

Zhengzhou
547
3
5049
9
0.017
8
0.643
3

Xian
543
4
3831
16
0.022
7
0.641
4

Hangzhou
526
5
5499
7
0.012
16
0.634
7

Tianjin
524
6
6092
3
0.013
13
0.636
5

Wuhan
521
7
5710
5
0.009
24
0.635
6

Nanchang
521
7
3718
17
0.010
21
0.632
9

Xuzhou
520
9
4254
14
0.010
22
0.631
10

Guangzhou
516
10
5209
8
0.026
3
0.634
8

Nanjing
509
11
6395
2
0.010
20
0.628
13

Shenyang
507
12
5610
6
0.023
4
0.631
11

Shijiazhuang
499
13
3475
19
0.015
11
0.629
12

Jinan
495
14
3985
15
0.011
19
0.622
14

Jinzhou
463
15
2523
36
0.009
26
0.614
16

Bengbu
461
16
2629
32
0.007
36
0.611
19

Qinhuangdao
461
16
5011
10
0.008
32
0.612
17

Suzhou
461
16
3586
18
0.008
33
0.612
18

Shangqiu
420
19
2695
30
0.006
40
0.614
15

Zhuzhou
451
20
2635
31
0.006
46
0.610
20

(a)

(b)

(c)

Figure 6: (a) The relationship between nodal degree and strength in CRN. (b) The relationship between nodal degree and betweenness in CRN. (c) The relationship between nodal degree and closeness in CRN.

Nodal degree is positively correlated with betweenness and closeness (see Figures 6(b) and 6(c)), which can be fitted by a power function and a linear function (for degree ≥ 100), 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 [5], Internet networks, and social networks [25], nodal degree and betweenness have a strong linear relation. In contrast, CRN shows some anomalies: certain cities have small degree and large betweenness (circled by red ellipses in Figure 6(b)).

To explore the underlying reasons for anomalies, cities are classified into five tiers based on nodal degree and betweenness, respectively (see Figures 7(a) and 7(b)). Prominent differences can be identified in the two maps. These differences identified between the two classifications indicate the strong influences 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 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. Thus, they play critical roles in connecting small cities scattered around them to other cities in the network.

(a)

(b)

Figure 7: (a) The spatial distribution of cities classified by nodal degree value in descending order. (b) The spatial distribution of the cities classified by nodal betweenness value in descending order.

5. The Role of Cities in CRN

Centrality measures, including degree, strength, betweenness, and closeness, reflect different aspects of cities’ importance 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. That 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. The 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 (0.033), but its connectivity (396) is weaker than that of Jinzhou (463), whose betweenness is only 0.009. 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. [26] 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 define it as a hub indicator to quantify the global roles of cities in the network. This measure is as follows:

where α, β, γ, and δ are respective weights of the unified nodal degree, strength, betweenness, and closeness. The values of α, β, γ, and δ are calculated using the coefficient of variation method, a data-driven method for measuring quantity weights (see the references for the details on the procedures for the coefficient of variation method) [27, 28]., , , and are calculated to be 0.22, 0.14, 0.56, and 0.08, respectively, and is in the range of 0.001 to 0.98 with an average value of 0.28.

Following the classification rules suggested by Guimerá et al. [5] and Du et al. [4], we divided the cities into four categories based on the value of via the k-means clustering algorithm: national core cities with , bridge cities with , peripheral cities with , and ultraperipheral cities with . The spatial distribution of the city categorizations is plotted in Figure 8. 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. This categorization is significant and consistent with the organization and evolution of CRN.

Figure 8: The spatial distribution of the cities in different categories according to the proposed integrated index.

There are 33 cities and 524 edges (0.8% of the total edges) in the core layer, 158 cities and 8,023 edges (11.9%) in the bridge layer, 451 cities and 10,838 edges (16.0%) in the peripheral layer, and 550 cities and 2,095 edges (3.1%) in the ultraperipheral layer. Moreover, there are 4,752 edges (7.1%) between the core layer and the bridge layer; 9,830 edges (14.5%) between the core layer and the peripheral or ultraperipheral layer; 25,490 edges (37.7%) between the bridge layer and the peripheral or ultraperipheral layer; and 6,024 edges (8.9%) between the peripheral layer and the ultraperipheral layer. A remarkable finding from this analysis is that stable connections hold between different layers in CRN, which is substantially different from the Chinese airline network, where most connections (63%) are within the core layer and minimal connections (0.25%) exist between the core layer and the peripheral layer [4]. This finding further demonstrates that CRN is a mature and efficient infrastructure network.

6. Conclusion

We investigated CRN by modeling it as a complex weighted network. Our findings suggest that CRN is a small-world, scale-free infrastructure network with a small average path length (2.21) and a large cluster coefficient (0.68). Unlike other complex networks such as the Internet and social and biological networks, the distributions of nodal centralities of 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 influences of social, political, and geographical factors.

Train traffic 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 findings show that uneven population distributions and economic clout account for the uneven distribution of CRN services between southeast and northwest China. The “four vertical and four horizontal” railway corridors establish major connections and train flows of CRN and pass through cities that are national or local political centers with advanced economies and dense populations. This indicates that the uneven distribution of CRN services reflects a strong political influence.

Nodal degree, strength, betweenness, and closeness quantify the importance of cities in CRN from different perspectives. However, no single measure can uncover the role of cities on a global scale. Thus, we proposed an integrated indicator that reveals the multilayer structure of CRN. It classified cities into four categories (core cities, bridge cities, peripheral cities, and ultraperipheral cities). Unlike the Chinese airline network, CRN has remarkably stable connections between different layers of the network, demonstrating the CRN’s accessibility and efficiency.

Our research has some limitations. China’s railway transportation system comprises different types of trains, such as G-number, D-number, C-number, Z-number, T-number, and K-number, with varying speeds and capacities. This research focuses on the connectivity of CRN and represents trains between cities as weighted edges in the topological network without considering train type. Assigning different weights to different trains will enable a more comprehensive analysis but requires substantial efforts. This is now included in our agenda for future research. Another direction is to examine networks formed by different 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

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

The authors are ordered alphabetically.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants No. 71490722, 71802166, and 41471383).

References

J. Berechman, D. Ozmen, and K. Ozbay, “Empirical analysis of transportation investment and economic development at state, county and municipality levels,” Transportation, vol. 33, no. 6, pp. 537–551, 2006.

Y. T. Mohmand, A. Wang, and A. Saeed, “The impact of transportation infrastructure on economic growth: empirical evidence from Pakistan,” Transportation Letters, vol. 9, no. 2, pp. 63–69, 2016.

M. Alderighi, A. Cento, P. Nijkamp, and P. Rietveld, “Network competition - The coexistence of hub-and-spoke and point-to-point systems,” Journal of Air Transport Management, vol. 11, no. 5, pp. 328–334, 2005.

W.-B. Du, X.-L. Zhou, O. Lordan, Z. Wang, C. Zhao, and Y.-B. Zhu, “Analysis of the Chinese Airline Network as multi-layer networks,” Transportation Research Part E: Logistics and Transportation Review, vol. 89, pp. 108–116, 2016.

R. Guimerà, S. Mossa, A. Turtschi, and L. N. Amaral, “The worldwide air transportation network: anomalous centrality, community structure, and cities' global roles,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 102, no. 22, pp. 7794–7799, 2005.

T. Verma, N. A. M. Araújo, and H. J. Herrmann, “Revealing the structure of the world airline network,” Scientific Reports, vol. 4, article 5638, 2014.

C. Liu, J. Wang, H. Zhang, and M. Yin, “Mapping the hierarchical structure of the global shipping network by weighted ego network analysis,” International Journal of Shipping and Transport Logistics, vol. 10, no. 1, pp. 63–86, 2018.

J. Feng, X. Li, B. Mao, Q. Xu, and Y. Bai, “Weighted complex network analysis of the Beijing subway system: Train and passenger flows,” Physica A: Statistical Mechanics and its Applications, vol. 474, pp. 213–223, 2017.

G. Riccardo and B. Marc, “Anatomy and efficiency of urban multimodal mobility,” Scientific Reports, vol. 4, no. 1, pp. 6911–6911, 2014.

X.-H. Yang, G. Chen, S.-Y. Chen, W.-L. Wang, and L. Wang, “Study on some bus transport networks in China with considering spatial characteristics,” Transportation Research Part A: Policy and Practice, vol. 69, pp. 1–10, 2014.

W. Li and X. Cai, “Empirical analysis of a scale-free railway network in China,” Physica A: Statistical Mechanics and its Applications, vol. 382, no. 2, pp. 693–703, 2007.

P. Sen, S. Dasgupta, A. Chatterjee, P. A. Sreeram, G. Mukherjee, and S. S. Manna, “Small-world properties of the Indian railway network,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 67, no. 3, Article ID 036106, 5 pages, 2003.

H. Soh, S. Lim, T. Zhang et al., “Weighted complex network analysis of travel routes on the Singapore public transportation system,” Physica A: Statistical Mechanics and its Applications, vol. 389, no. 24, pp. 5852–5863, 2010.

D. Efthymiou, C. Antoniou, E. Siora, and D. Argialas, “Modeling the impact of large-scale transportation infrastructure development on land cover,” Transportation Letters, vol. 10, no. 1, pp. 26–42, 2016.

J. Jiao, J. Wang, and F. Jin, “Impacts of high-speed rail lines on the city network in China,” Journal of Transport Geography, vol. 60, pp. 257–266, 2017.

R. Wang, J.-X. Tan, X. Wang, D.-J. Wang, and X. Cai, “Geographic coarse graining analysis of the railway network of China,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 22, pp. 5639–5646, 2008.

Y. Wang, T. Zhou, J. Shi, J. Wang, and D. He, “Empirical analysis of dependence between stations in Chinese railway network,” Physica A: Statistical Mechanics and its Applications, vol. 388, no. 14, pp. 2949–2955, 2009.

S.-R. Zou, T. Zhou, A.-F. Liu, X.-L. Xu, and D.-R. He, “Topological relation of layered complex networks,” Physics Letters A, vol. 374, no. 43, pp. 4406–4410, 2010.

J. Lin and Y. Ban, “Complex network topology of transportation systems,” Transport Reviews, vol. 33, no. 6, pp. 658–685, 2013.

D. J. Watts and S. H. Strogatz, “Collective dynamics of 'small-world' networks,” Nature, vol. 393, no. 6684, pp. 440–442, 1998.

T. Opsahl, F. Agneessens, and J. Skvoretz, “Node centrality in weighted networks: Generalizing degree and shortest paths,” Social Networks, vol. 32, no. 3, pp. 245–251, 2010.

J. Wang, H. Mo, F. Wang, and F. Jin, “Exploring the network structure and nodal centrality of China's air transport network: a complex network approach,” Journal of Transport Geography, vol. 19, no. 4, pp. 712–721, 2011.

F. Cowell, Measuring Inequality, Oxford University Press, 2011.

Z. Xu and R. Harriss, “Exploring the structure of the U.S. intercity passenger air transportation network: A weighted complex network approach,” GeoJournal, vol. 73, no. 2, pp. 87–102, 2008.

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. W. Hwang, “Complex networks: Structure and dynamics,” Physics Reports, vol. 424, no. 4-5, pp. 175–308, 2006.

X. Luan, B. Yang, and Y. Zhang, “Structural hierarchy analysis of streets based on complex network theory,” Geomatics and Information Science of Wuhan University, vol. 37, no. 6, pp. 728–732, 2012.

H. Abdi, “Coefficient of variation,” Encyclopedia of Research Design, vol. 1, pp. 169–171, 2010.

A. G. Bedeian and K. W. Mossholder, “On the use of the coefficient of variation as a measure of diversity,” Organizational Research Methods, vol. 3, no. 3, pp. 285–297, 2000.