Comparison analysis on complex topological network models of urban rail transit: A case study of Shenzhen Metro in China

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Abstract

To study the topological complexity of urban rail transit (URT) networks with the multi-line transfer stations from different perspectives, Shenzhen Metro (SZM) is taken as an example and Space L & Space P models are established in this study. Then, based on multiple evaluation parameters and key nodes ranking, the differences of network topological complexity in two models are deeply explored and compared quantitatively. Some meaningful results have been obtained: (i) The characteristics of scale-free networks in Space L and Space P are proved through the eigenvector centrality distribution and truncated power-law distribution of cumulative degree. Scale-free networks show both robustness against random faults and vulnerability against deliberate attacks. The daily safety management at 16.87% of hub stations in Space P and 17.47% of hub stations in Space L should be taken seriously by metro managers in case of emergency events. (ii) Since the WS small-world effect in Space P model is more evident than that in Space L model, the connections among stations and OD accessibility of passenger are enhanced in Space P network. (iii) The important and risk nodes are concentrated in Space L and are more decentralized in Space P. P model has the stronger overall anti-attack capability than L model, which is more beneficial to the resilience of network. This study can realize the deeper understanding of URT system with different models and it can provide theoretical support for complex network analysis of URT system.

Introduction

Network and relationships are interdependent from the perspective of system dynamics. Nowadays, network comes into being in many aspects of nature, such as public transportation system [1], [2], worldwide air [3], [4] or marine [5] transport network, power grid system [6], social media network [7], ecological systems [8], neural network [9], network traffic flows [10] and other information transmission processes. With the development of science and technology, network becomes more and more complex. Therefore, complex network theory is very important when we are faced with these problems. Complex network is composed of many nodes and the edges connected between them. The nodes represent the components of the system, while the edges describe the interaction between two nodes. Strogatz summarized that the complexity of the current network was mainly concentrated on six aspects, including the structural complexity, network evolution, connection diversity, dynamical complexity, node diversity and meta-complication [11].

Heretofore, the studies on the application of complex network theory to urban public transportation, especially urban rail transit system, have been developing vigorously [12], [13]. To sum up, most of them analyze the local and overall characteristics of the network from different calculation indexes based on complex network modeling, including average degree, average shortest path length, degree distribution, clustering coefficient, network entropy, etc. Besides, some studies are carried out to optimize the evaluation parameters or further analyze the networks, mainly involving the centrality of metro network [12], integrated indicators [14], simulation of dynamic evolution trend [13], [15] and passenger flow weighted method [2], [16], [17]. Moreover, heterogeneity [16], vulnerability [16], [18], robustness [19], [20] or resilience [21] of metro network were analyzed.

However, the first step to determine the characteristics of rail transit network is to establish a scientific analysis model. Taylor [22] described four topological models of public transportation network in Chapter Six of his book, including Space-L, Space-B, Space-P and Space-C. Fig. 1 depicts an unvalued and undirected metro network consisting of four lines from different perspectives. Among them, the representation of Space L model is a schematic physical or geographic layout of the transit network. Each station represents a node. If two nodes are two consecutive stations on one or multiple transit lines, they will be connected with an edge. In Space B model, all lines and stations are nodes. Each line node is connected to each station node on it and nodes of the same kind are not connected. Moreover, in Space P model, all stations constitute network nodes. If any two nodes arrive directly via the same line or are accessible by transferring in multiple lines, they will have one connecting edge. At last, the Space C model is the simplest compared with other three models. All lines are network nodes, and if two lines are connected through a transfer station, they have a link. Comparing the four models, Space C is the easiest and Space P is the most complex. The adjacent relations between nodes are emphasized in Space L, while Space P highlights the transfer characteristics. Space C omits the stations and shows the largest difference from the real transit network. Although Space B shows all the stations and lines, it ignores the mutual relationship between the stations. In short, Space L model and Space P model are relatively closer to the real rail transit network. Therefore, this study will focus on the modeling and comparative analysis of the two networks in URT.

In recent years, the construction of urban rail transit has been developing more and more prosperous and vigorous in China, with the more and more developed and complex metro network. By the end of 2018, 35 cities in the Chinese mainland had opened urban rail transit systems with 185 lines in operation and a total length of 5,761.4 kilometers [23]. The crisscrossing lines constitute the metro topological network, which is equivalent to the skeleton of urban rail transit. Therefore, the topological structure of metro is the fundamental reason for the distribution of passenger flow and it is very important to explore the complex topological networks of rail transit. However, most of the rail transit network modeling is based on line network schematic diagram, namely Space L model, and further researches on multi-space modeling of URT networks are needed.

To study the topological complexity of URT networks with the multi-line transfer stations from different perspectives, Shenzhen Metro (SZM) system with a four-line transfer station is selected as the research object. Space L and Space P network models based on SZM are built and the differences of network topological complexity are analyzed based on six evaluation parameters. Finally, the relations between parameters are discussed through factor analysis method and the weighted importance ranking of all nodes is visualized and concluded. This study can help us understand URT system with different network models better.

Section snippets

Network modeling of Shenzhen metro

Up to December 31, 2018, SZM network structure currently consists of eight lines (L-1/2/3/4/5/7/9/11) and 166 stations, including 28 transfer stations. Here is the rule for numbering the stations. According to the order of L-1/2/3/4/5/7/9/11, we can number the stations in all lines. Thus, LH Station is numbered 1, LJ Station is 3 and AE Station is 30. Especially, for those transfer stations, the codes are the numbers with the smaller line. In this way, the overall network can be digitally coded.

Comparison analysis results

The two network models of Space L and Space P can be described with some basic parameters in Table 1. It can be acquired that for 166 nodes in SZM network, the number of edges in Space P is nearly 13 times as much as that in Space L, which is determined by the specific properties of the two networks. Therefore, the network diameter varies greatly in two networks and the average degree is the same reason. The furthest distance in Space L network is 43 from SL to BT station and it is 3 in P

Relationship between three centralities

Through the statistical analysis of degree, betweenness, closeness and eigenvector centrality of all nodes in Space L and Space P, it can be deduced that there are very close mathematical relationships between the four node centralities in P model. As shown in Fig. 9(d), the relationships among the four centralities in Space L have obvious fluctuation characteristics. To better demonstrate the mathematical relationships in P model, we give the contour map distributions between four

Conclusions

Taking Shenzhen Metro system as an example and based on the topological theories of complex network, Space L and Space P which respectively highlighted the adjacent and transferring characteristics are modeled. Six classical evaluation parameters in the two models are presented and compared. The important station ranking in two models is obtained with factor analysis method. Some conclusions can be drawn:

(1) From the comparative analysis of degree centrality, the degree distributions of nodes

CRediT authorship contribution statement

Yangyang Meng: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Xiangliang Tian: Visualization, Reviewing. Zhongwen Li: Resources, Funding acquisition. Wei Zhou: Investigation, Funding acquisition. Zhijie Zhou: Investigation, Funding acquisition. Maohua Zhong: Reviewing, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

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

Acknowledgments

This work was supported by the National Key R&D Program of China (Grant No. 2018YFC0809900 and 2016YFC0802500) and Postdoctoral Innovative Talents Support Program, China (Grant No. BX20180158), We thank editor and reviewers for the valuable comments.

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