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Identifying hub stations and important lines of bus networks: A case study in Xiamen, China
Accepted Manuscript Identifying hub stations and important lines of bus networks: A case study in Xiamen, China Hui Zhang, Chengxiang Zhuge, Xiaohua Yu
PII: DOI: Reference:
S0378-4371(18)30274-7 https://doi.org/10.1016/j.physa.2018.02.182 PHYSA 19302
To appear in:
Physica A
Received date : 23 April 2017 Revised date : 17 November 2017 Please cite this article as: H. Zhang, C. Zhuge, X. Yu, Identifying hub stations and important lines of bus networks: A case study in Xiamen, China, Physica A (2018), https://doi.org/10.1016/j.physa.2018.02.182 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Propose two new indicators-neighborhood degree ratio and transfer index to evaluate nodes in bus networks. Transfer stability has been tested by an introduced node failing process. Deliberate attack by transfer index of high value can cause more damages. Important lines have been found according to line transfer accessibility.
*Manuscript Click here to view linked References
Identifying hub stations and important lines of bus networks: A case study in Xiamen, China Hui Zhanga,*, Chengxiang Zhugeb, Xiaohua Yua a. School of Transportation Engineering, Shandong Jianzhu University, Jinan, 250101, China b. Department of Geography, University of Cambridge, Cambridge CB2 3EN, UK
Abstract: Hub stations and important lines play key roles in transfers between stations. In this paper, a node failure model is proposed to identify hub stations. In the model, we introduce two new indicators called neighborhood degree ratio and transfer index to evaluate the importance of stations, which consider neighborhood stations’ degree of station and the initial transfer times between stations. Moreover, line accessibility is developed to measure the importance of lines in the bus network. Xiamen bus network in 2016 is utilized to test the model. The results show that the two introduced indicators are more effective to identify hub stations compared with traditional complex network indicators such as degree, clustering coefficient and betweenness.
Keywords: Hub stations, Bus networks, Transfer index, Node failing process 1. Introduction Urban bus system has been widely considered to be an effective way in alleviating traffic congestion in cities. With the rapid development of urbanization, bus networks have become too complicated to evaluate. Bus networks are typical complex networks with high values of clustering coefficient and comparatively low mean shortest path values [1], which are the fundamental components in bus systems. Complex network theory provides a powerful tool to understand the functions and mechanism of transportation network, and it has been applied to many transport networks such as bus, subway, airport and railway [2-6]. There are numerous indicators to measure network characters in the past two decades including graph indicators and complex network indicators such as degree, clustering coefficient, betweennes, efficiency, community and correlated degree [7-10]. Stations and lines are two basic components of a bus network. The hub nodes and important lines are more influential for transfers and traffic dynamics. In complex systems, identifying these nodes is conducive to protect network stability to avoid vulnerability. Chen proposed three metrics to assess system vulnerability and identified critical nodes [11]. Zhang et al. studied the characteristics on hub networks of urban transit network, and the results show the transfer station corresponding to
*Corresponding author
E-mail address: [email protected] (H. Zhang)
route plays the most important role [12]. Sun et al. used two indicators of z-score and participation coefficient to measure the influence of nodes in Dublin bus network [13]. In reality, bus networks often suffer from interruption in service including traffic congestion, road work zones and bad weather. Removal of these nodes or lines decreases the ability of network to transport passengers, which is a useful method to test network stability. Researches revealed that intentional attacks could cause severe collapse of large networks [14-16]. Pang et al. studied the efficiency and robustness of different bus network design and found that dual-blob type is least robust with targeted node removal but is most resilient with random node failures [17]. Ren et al. studied the robustness of Shenyang bus network with three static network models, they found that the network was robust under random attack but more fragile under the selective attack [18]. There are also a few researches focusing on the role of bus lines in network [9, 19]. Transfers play vital roles in connecting different lines in bus networks. There are many efforts focusing on transfer coordination in transportation area [20, 21]. The average transfer time of a network is a significant indicator to assess performance of a transit network. High effective transit network require small value because transfers enhance passengers’ travel time and bring inconvenience. Space P and space L are two essential methods to build complex network model [1, 22]. Space P is utilized to calculate transfers. Zhang et al. proposed an extremely simple method to calculate transfer time between two nodes in bus networks using Space P [9]. While space L is used to measure the basic topological characteristics of complex network. Previous exploratory researches show that the complex bus networks are scale-free networks and the nodes with large degree play more important role in the network [23-25]. There are many researches that focused on analyzing the properties of bus networks. However, these models ignore the relationship between network structure and transfers, especially the mechanism of transfer construction. Limited work has been done to analyze the roles of bus stations and lines. Moreover, it remains a lack of understanding how the hub nodes and important lines impact the accessibility of a bus network. To fill this gap, this paper proposes a node failure process to identify the hub nodes considering transfer changes. Line accessibility will be introduced to measure the role of lines in the network. The reminder of this paper is organized as follows. Section 2 introduces the background of bus networks. Section 3 gives the data description. Section 4 presents the hub nodes identification. Section 5 introduces the important line identification. Conclusions are given in section 6.
2. Background 2.1Bus network representation Urban bus network is a typical complex network that contains stations and lines. Two different lines are connected with each other at transfer stations. We use space L and space P to represent a bus network respectively. Space L and space P are two famous spaces that have been well introduced in the former studies [1, 2]. The topological bus network of space L contains nodes which represent bus stations, and
there is an edge between two nodes if they are consecutive stations on the route. The topology of space L is always used to analyze the various properties of bus network. Nodes in space P are the same as in space L. While an edge between two nodes means there is a line passing through them. The bus network in space P is also called transfer network, which is useful to analyze the transfer property. Fig.1 shows a bus network under space L and space P. In this paper, a bus network can be represented as a graph G (V , E ) , V is the set of nodes, E is the set of edges. G is the N N adjacency matrix {eij } . If there exists an edge between node i and node j , eij 1 ; otherwise eij 0 . N is the number of bus stations in the network. 6
1
2
13
12
3
4
8
10
11
11
1
5
6
2
7
8
12
5
13
9
7
3
(a)
4 (b)
10
9
Fig. 1. Representations of a bus network: (a) space L and (b) space P.
2.2. Transfer times calculation Transfer time reflects the accessibility between stations. So, the average transfer time of a bus network is an important indicator to evaluate its performance. It is complicated to calculate transfer time before the emergence of complex network theory. In the former work, a method has been developed to calculate the average transfer times using Floyd algorithm under space P [23]. Fig. 1(b) shows there are three lines (red dashed) between node 6 and node 13, which means there is two transfer times between them. The smallest transfer time sttij between node i and node j can be expressed as sttij sijP 1, i j
(1)
P where sij is the shortest path between node i and node j . So, the average transfer
time of a bus network can be defined as N
N
stt Atr=
i 1 j=1
ij
N ( N 1)
,i j
(2)
3. Data description The data used in this paper is from the bus system of Xiamen that consists of 712
lines (including upstream and downstream) and 2163 stations. The raw data contains the names of stations and lines, the number of lines and stations, the latitude and longitude of stations. Fig. 2 shows Xiamen bus network layouts including real network, topological structures under space L and space P.
Fig. 2. Xiamen bus networks: (a) real network, (b) topological structure under space L and (c) topological structure under space P.
4. Hub nodes identification It is known that transit networks are very fragile to deliberate attacks to some nodes with large degree and betweenness [26]. In bus networks, nodes failures can cut off transfer lines, which results in low effectiveness of whole networks. In reality, transfer nodes failures will bring lots of troubles when they happen in rush hours. Traffic designers and managers pay more attentions to those important nodes. Obviously, nodes with large values of degree, betweennes, clustering coefficient plays more crucial roles in networks. However, it is not clear that is there any nodes play more important roles in a bus network. In this paper, we define hub nodes are the ones having significant impacts on transfer efficiency. Generally, the hub nodes in complex networks mean the nodes with high degree, betweenness or clustering coefficient. In this paper, the relationship between transfer times in bus networks and those indicators will be studied. The situation of degree, betweenness and clustering coefficient of Xiamen under space L are introduced as follows. (1) Degree. The degree k i of node i is defined as the number of nodes that connected with the node i . Degree contains in-degree and out-degree and it follows exponential or power law degree distribution in most public transport networks [27]. Degree is a crucial indicator to measure the centrality of a node in the network. It is calculated the degree interval of Xiamen stations is k [1, 25] and the average degree k 3.8 . Fig. 3 shows the degree distribution of Xiamen bus network in double
logarithmic scale, and it follows a shifted power law degree distribution f (k ) 7.24* (k ) / (k 2.8) . The horizontal coordinate is the value of degree, and
the vertical coordinate is the proportion. The results show that the stations with k 2 account for 41% and the stations with k 10 account for 2.87%, which indicates
there are only a few hub stations connected with many links.
Fig. 3. Degree distribution of Xiamen bus network in double logarithmic scale.
(2) Betweenness. Betweenness is also an indicator to measure the centrality of a node in a network. It is defined as [28]
st (v) st s t
C B (v )
(3)
where st is the number of shortest paths going from s to t and st (v) is the number of shortest path going form s to t through the node v . Fig. 4 exhibits the betweenness values of nodes with rank in double logarithmic scale. The horizontal coordinate is the node number, and the vertical coordinate is the betweenness value. The value is CB v [0,0.1334] and the average value is 0.0011. It is calculated that only 2.2% stations with CB v 0.02 .
Fig. 4. Betweenness values of nodes of Xiamen bus network in double logarithmic scale.
(3) Clustering coefficient. Clustering coefficient is an important property of characterizing the local cohesiveness of the current node or the extent to which the nodes in the network are clustered together. It can be defined as C (vi )
2ei mi (mi 1)
(4)
where ei is the actual number of edges shared with local neighbors of node vi , mi is the connection degree of neighbors of node vi . Fig. 5 shows the values of clustering coefficient in Xiamen bus network. The horizontal coordinate is the node number. The values are distributed from 0 to 1 and the average value is 0.18. The stations with value C (vi ) 0.7 account for 6.3%.
Clustering coefficient
1 0.8 0.6 0.4 0.2 0 0
500
1000 1500 2000 2500 Node Fig. 5. Values of clustering coefficient of nodes sorted in Xiamen bus network.
4.1 Proposed indicators In networks, nodes have tendency to connect other nodes with similar type is called assortative networks, while nodes’ tendency to connect other nodes with different type is called disassortative networks [29]. The degree correlation plays an important role in traffic dynamics and self-organized criticality [30, 31]. Moreover, initial transfer times between nodes are also very important in a bus network. In this paper, we propose two indicators to evaluate the importance of nodes: neighborhood degree ratio and transfer index. (1) Neighborhood degree ratio. Neighborhood degree ratio i is defined as
i =
ki kj
Ni
where k i is the degree of the node i , N i is the set of neighbors of node i , and
(5)
kj
Ni
is the average degree of its neighbors. The large the value is, the important the
node is. Fig.6 exhibits the values of neighborhood degree ratio in Xiamen network ranging from 0.0175 to 2.9091. It is calculated that the average neighborhood degree ratio is 0.81. The stations with i 2 account for 1.85%. 3 2.5
i
2 1.5 1 0.5 0 0
500
1000 1500 Node
2000
2500
Fig. 6. Neighborhood degree ratios of nodes sorted in Xiamen bus network.
(2) Transfer index i . Node transfer index reflects the ability of accessibility to other nodes. It can be defined as N
i = ( j=1
1
)/ N stt ij 1
(6)
where sttij is the smallest transfer time from node i and node j . According to the definition, the element 0 i 1 . Large value means high ability of accessibility. Fig. 7 shows the values of node transfer index in Xiamen bus network sorted from 0.2828 to 0.6197. The average value is 0.42, and the stations with i 0.55 account for 2.6%。
0.65 0.6 0.55
i
0.5 0.45 0.4 0.35 0.3 0.25 0
500
1000 1500 Node
2000
2500
Fig. 7. Values of node transfer efficient sorted in Xiamen bus network.
4.2 Node failing process In this paper, we propose a node failing process to identify the hub nodes in bus networks. The process is called deliberate attack. When a node is attacked, all the edges that connected with the node under space P will be removed. Once the node has been attacked, the transfer condition of network will change. We use to represent the proportion of zero and one transfer time in a bus network, which can reflect the network transfer efficiency. There are six strategies to attack nodes in network deliberately according to degree, clustering coefficient, betweenness, neighborhood degree ratio, transfer index and random choice. There are two steps of the process. Take degree for example, (ⅰ) sorting the nodes with degree from large to small firstly, (ⅱ) attacking nodes step by step. 0.44
0.42
0.4
0.38
0.36
0.34
0.32 0
Transfer index random Neighborhood degree ratio Clustering coefficient Degree Betweenness 10
20
30
40
50
Step Fig. 8. Proportions of zero and one transfer time in Xiamen bus network with 50 different node
failing processes.
Fig. 8 illustrates the proportions of zero and one transfer ( ) in Xiamen bus network with 50 different node failing processes. The horizontal coordinate is the failing step, the vertical coordinate is the proportion of zero and one transfer time of the bus network. As can be seen, the decreases as the nodes been attacked. Take degree for example, are 0.4303 , 0.4193, 0.4112, 0.4020, 0.3883 and 0.3696 at step 0, 10, 20, 30, 40 and 50 respectively. Moreover, it is found that attacking nodes with large transfer index is more effective to destroy the network compared with other indicators. It is can be seen in the figure, are 0.3376, 0.3626, 0.3696, 0.3848, 0.4118 and 0.4133 after 50 steps for transfer index, neighborhood degree ratio, degree, betweenness, random and clustering coefficient respectively. In addition, the results also indicate that the bus network is robust by large clustering coefficient and random attack. To better understand the impact of attack on transfer time of the bus network, this paper investigates the proportions of zero, one, two, three, four and more transfer times after each attack. Fig. 9 shows the proportions by node failing process with large transfer index. It can be noted that the proportions of zero, one and two transfer times decrease as the nodes failing. For example, the proportions of two transfer times are 0.5170, 0.4999, 0.4944, 0.4931, 0.4902 and 0.4892 after failing 0, 10, 20, 30, 40 and 50 nodes respectively. While the proportions of three and more transfer time increase as the nodes failing. For example, the proportions of four times and more are 0.0051, 0.0161, 0.0264, 0.0357, 0.0472 and 0.0563 after failing 0, 10, 20, 30, 40 and 50 nodes respectively. 0.7 0.6
Proportion
0.5 0.4 0.3 0.2
Zero Transfer One transfer Two transfers Three transfers Four transfers and more
0.1 0 0
10
20
30
40
50
Step
Fig. 9. Proportion of zero, one, two, three, four and more transfer times in Xiamen bus network after node failing with large transfer index.
Table 1 shows the top 30 stations of Xiamen bus network with large transfer index. Station with large value means it is more important for network transfer efficiency. Failures of these stations can result in numerous unnecessary transfers, which enhance the total transfer time of the whole network. These stations are called hub stations in the bus network. Traffic managers should protect the traffic environment to enhance the efficiency of the bus network. Moreover, hub stations are also very useful in bus
network optimization. Table 1. Top 30 stations of Xiamen bus network with large transfer index. No.
Name
Value
No.
Name
Value
No.
Name
Value
1
SM Square
1340.3
11
Sibei Lukou
1249.0
21
Huoche Zhan
1231.3
2
Xingbei
1282.3
12
Zhaohu Zhan
1243.3
22
Haicang
1227.3
3
Banlian Guomao
1265.2
13
Chengxi
1241.0
23
Qiche Zhan
1225.3
4
Kaihe Lukou
1263.8
14
Dongdu
1240.5
24
Haiyu Lu
1224.3
5
Dianqian
1263.2
15
Lvcuo
1240.5
25
HC Jianhang
1224.3
6
Xiamen Bei
1258.8
16
Gaoqi Zhan
1240.2
26
Xilin
1224.0
7
Malong
1258.7
17
Shangjian
1239.0
27
Lianban
1222.7
8
Tangbian
1258.0
18
Lianhu Lukou
1234.8
28
Xinglin
1222.7
9
Xiaoxi Men
1253.7
19
Chengnan
1232.2
29
Huagong
1222.7
10
Neimao
1252.7
20
Xiangfeng
1231.7
30
Qinggong
1221.7
5. Identification of important lines Bus lines determine the transfer modes in a bus network. As the network become increasingly large, some lines play more and more important roles in transfers, which enhance the efficiency of bus networks. Line evaluation is crucial for bus network design. In this paper, we use transfer accessibility to measure the importance of lines in a bus network. We firstly define the line transfer matrix as follows: Tmm tij
i j tij ij i, j 1, 2, 0 i j
(7)
,m (8)
where Tmm is the line transfer matrix, m is the number of lines, ij is the number of transfer nodes between node i and node j . So, line transfer accessibility i of line i can be defined as m
i tij
(9)
j 1
Obviously, the large value of line transfer accessibility means less transfer time to other lines. Fig. 10 gives a simple example of a bus network with four lines. There are four transfer nodes in the network. Table 2 shows the line transfer matrix of the network. According to formula (9), a 2 , b 3 , c 2 , d 1 . The results indicate that line b is an important line from the perspective of transfers.
b a
c
d
Fig. 10. A simple example of a bus network with four lines. Table 2. The line transfer matrix of the simple example. a b c d
Line No.
Summation
a
0
1
1
0
2
b
1
0
1
1
3
c
1
1
0
0
2
d
0
1
0
0
1
Summation
2
3
2
1
0
Fig. 11 illustrates the line transfer accessibility for all lines. The horizontal coordinate is the line number. It is can be seen that the values of are from 2 to 1609. Moreover, there are only 3.23% of lines with 1000 , which contains lines ‘651’, ‘659’, ‘655’, ‘792’, ‘791’, ‘455’ et al. This means these lines with large play more important roles in network accessibility. 1800 1600 1400 1200
1000 800 600 400 200 0 0
100
200
300
400 Line
500
600
700
800
Fig. 11. Line transfer accessibility for all lines sorted.
6. Conclusions This paper constructed two complex network models by space L and space P. Space L is used to calculate the basic complex network indicators such as degree, clustering coefficient and betweenness. Space P is used to calculate transfer times between nodes. Neighborhood degree ratio and transfer index have been proposed to measure
the importance of stations. Moreover, a node failure process has been developed to identify hub stations in the bus network. Xiamen bus network is used to test the node failure model. The results show that the proposed transfer index is more effective to identify hub station in a bus network compared with other indicators. In addition, line transfer accessibility has been proposed to measure the importance of lines. The general idea of identification of hub stations and important lines has potential application in urban transit network design and management. In the future research, more cities’ bus network would be collected to verify the accuracy of proposed model. The model will consider the real transit demand combined with network structure. In addition, dynamic evaluation of urban transit network is another interesting research.
Acknowledgement The Project was supported by the Doctoral funds of Shandong Jianzhu University (XNBS1614), open fund for the key laboratory for traffic and transportation security of Jiangsu province (TTS2016-02) and Shandong Provincial Natural Science Foundation (ZR2013EEQ014).
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PII: DOI: Reference:
S0378-4371(18)30274-7 https://doi.org/10.1016/j.physa.2018.02.182 PHYSA 19302
To appear in:
Physica A
Received date : 23 April 2017 Revised date : 17 November 2017 Please cite this article as: H. Zhang, C. Zhuge, X. Yu, Identifying hub stations and important lines of bus networks: A case study in Xiamen, China, Physica A (2018), https://doi.org/10.1016/j.physa.2018.02.182 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Propose two new indicators-neighborhood degree ratio and transfer index to evaluate nodes in bus networks. Transfer stability has been tested by an introduced node failing process. Deliberate attack by transfer index of high value can cause more damages. Important lines have been found according to line transfer accessibility.
*Manuscript Click here to view linked References
Identifying hub stations and important lines of bus networks: A case study in Xiamen, China Hui Zhanga,*, Chengxiang Zhugeb, Xiaohua Yua a. School of Transportation Engineering, Shandong Jianzhu University, Jinan, 250101, China b. Department of Geography, University of Cambridge, Cambridge CB2 3EN, UK
Abstract: Hub stations and important lines play key roles in transfers between stations. In this paper, a node failure model is proposed to identify hub stations. In the model, we introduce two new indicators called neighborhood degree ratio and transfer index to evaluate the importance of stations, which consider neighborhood stations’ degree of station and the initial transfer times between stations. Moreover, line accessibility is developed to measure the importance of lines in the bus network. Xiamen bus network in 2016 is utilized to test the model. The results show that the two introduced indicators are more effective to identify hub stations compared with traditional complex network indicators such as degree, clustering coefficient and betweenness.
Keywords: Hub stations, Bus networks, Transfer index, Node failing process 1. Introduction Urban bus system has been widely considered to be an effective way in alleviating traffic congestion in cities. With the rapid development of urbanization, bus networks have become too complicated to evaluate. Bus networks are typical complex networks with high values of clustering coefficient and comparatively low mean shortest path values [1], which are the fundamental components in bus systems. Complex network theory provides a powerful tool to understand the functions and mechanism of transportation network, and it has been applied to many transport networks such as bus, subway, airport and railway [2-6]. There are numerous indicators to measure network characters in the past two decades including graph indicators and complex network indicators such as degree, clustering coefficient, betweennes, efficiency, community and correlated degree [7-10]. Stations and lines are two basic components of a bus network. The hub nodes and important lines are more influential for transfers and traffic dynamics. In complex systems, identifying these nodes is conducive to protect network stability to avoid vulnerability. Chen proposed three metrics to assess system vulnerability and identified critical nodes [11]. Zhang et al. studied the characteristics on hub networks of urban transit network, and the results show the transfer station corresponding to
*Corresponding author
E-mail address: [email protected] (H. Zhang)
route plays the most important role [12]. Sun et al. used two indicators of z-score and participation coefficient to measure the influence of nodes in Dublin bus network [13]. In reality, bus networks often suffer from interruption in service including traffic congestion, road work zones and bad weather. Removal of these nodes or lines decreases the ability of network to transport passengers, which is a useful method to test network stability. Researches revealed that intentional attacks could cause severe collapse of large networks [14-16]. Pang et al. studied the efficiency and robustness of different bus network design and found that dual-blob type is least robust with targeted node removal but is most resilient with random node failures [17]. Ren et al. studied the robustness of Shenyang bus network with three static network models, they found that the network was robust under random attack but more fragile under the selective attack [18]. There are also a few researches focusing on the role of bus lines in network [9, 19]. Transfers play vital roles in connecting different lines in bus networks. There are many efforts focusing on transfer coordination in transportation area [20, 21]. The average transfer time of a network is a significant indicator to assess performance of a transit network. High effective transit network require small value because transfers enhance passengers’ travel time and bring inconvenience. Space P and space L are two essential methods to build complex network model [1, 22]. Space P is utilized to calculate transfers. Zhang et al. proposed an extremely simple method to calculate transfer time between two nodes in bus networks using Space P [9]. While space L is used to measure the basic topological characteristics of complex network. Previous exploratory researches show that the complex bus networks are scale-free networks and the nodes with large degree play more important role in the network [23-25]. There are many researches that focused on analyzing the properties of bus networks. However, these models ignore the relationship between network structure and transfers, especially the mechanism of transfer construction. Limited work has been done to analyze the roles of bus stations and lines. Moreover, it remains a lack of understanding how the hub nodes and important lines impact the accessibility of a bus network. To fill this gap, this paper proposes a node failure process to identify the hub nodes considering transfer changes. Line accessibility will be introduced to measure the role of lines in the network. The reminder of this paper is organized as follows. Section 2 introduces the background of bus networks. Section 3 gives the data description. Section 4 presents the hub nodes identification. Section 5 introduces the important line identification. Conclusions are given in section 6.
2. Background 2.1Bus network representation Urban bus network is a typical complex network that contains stations and lines. Two different lines are connected with each other at transfer stations. We use space L and space P to represent a bus network respectively. Space L and space P are two famous spaces that have been well introduced in the former studies [1, 2]. The topological bus network of space L contains nodes which represent bus stations, and
there is an edge between two nodes if they are consecutive stations on the route. The topology of space L is always used to analyze the various properties of bus network. Nodes in space P are the same as in space L. While an edge between two nodes means there is a line passing through them. The bus network in space P is also called transfer network, which is useful to analyze the transfer property. Fig.1 shows a bus network under space L and space P. In this paper, a bus network can be represented as a graph G (V , E ) , V is the set of nodes, E is the set of edges. G is the N N adjacency matrix {eij } . If there exists an edge between node i and node j , eij 1 ; otherwise eij 0 . N is the number of bus stations in the network. 6
1
2
13
12
3
4
8
10
11
11
1
5
6
2
7
8
12
5
13
9
7
3
(a)
4 (b)
10
9
Fig. 1. Representations of a bus network: (a) space L and (b) space P.
2.2. Transfer times calculation Transfer time reflects the accessibility between stations. So, the average transfer time of a bus network is an important indicator to evaluate its performance. It is complicated to calculate transfer time before the emergence of complex network theory. In the former work, a method has been developed to calculate the average transfer times using Floyd algorithm under space P [23]. Fig. 1(b) shows there are three lines (red dashed) between node 6 and node 13, which means there is two transfer times between them. The smallest transfer time sttij between node i and node j can be expressed as sttij sijP 1, i j
(1)
P where sij is the shortest path between node i and node j . So, the average transfer
time of a bus network can be defined as N
N
stt Atr=
i 1 j=1
ij
N ( N 1)
,i j
(2)
3. Data description The data used in this paper is from the bus system of Xiamen that consists of 712
lines (including upstream and downstream) and 2163 stations. The raw data contains the names of stations and lines, the number of lines and stations, the latitude and longitude of stations. Fig. 2 shows Xiamen bus network layouts including real network, topological structures under space L and space P.
Fig. 2. Xiamen bus networks: (a) real network, (b) topological structure under space L and (c) topological structure under space P.
4. Hub nodes identification It is known that transit networks are very fragile to deliberate attacks to some nodes with large degree and betweenness [26]. In bus networks, nodes failures can cut off transfer lines, which results in low effectiveness of whole networks. In reality, transfer nodes failures will bring lots of troubles when they happen in rush hours. Traffic designers and managers pay more attentions to those important nodes. Obviously, nodes with large values of degree, betweennes, clustering coefficient plays more crucial roles in networks. However, it is not clear that is there any nodes play more important roles in a bus network. In this paper, we define hub nodes are the ones having significant impacts on transfer efficiency. Generally, the hub nodes in complex networks mean the nodes with high degree, betweenness or clustering coefficient. In this paper, the relationship between transfer times in bus networks and those indicators will be studied. The situation of degree, betweenness and clustering coefficient of Xiamen under space L are introduced as follows. (1) Degree. The degree k i of node i is defined as the number of nodes that connected with the node i . Degree contains in-degree and out-degree and it follows exponential or power law degree distribution in most public transport networks [27]. Degree is a crucial indicator to measure the centrality of a node in the network. It is calculated the degree interval of Xiamen stations is k [1, 25] and the average degree k 3.8 . Fig. 3 shows the degree distribution of Xiamen bus network in double
logarithmic scale, and it follows a shifted power law degree distribution f (k ) 7.24* (k ) / (k 2.8) . The horizontal coordinate is the value of degree, and
the vertical coordinate is the proportion. The results show that the stations with k 2 account for 41% and the stations with k 10 account for 2.87%, which indicates
there are only a few hub stations connected with many links.
Fig. 3. Degree distribution of Xiamen bus network in double logarithmic scale.
(2) Betweenness. Betweenness is also an indicator to measure the centrality of a node in a network. It is defined as [28]
st (v) st s t
C B (v )
(3)
where st is the number of shortest paths going from s to t and st (v) is the number of shortest path going form s to t through the node v . Fig. 4 exhibits the betweenness values of nodes with rank in double logarithmic scale. The horizontal coordinate is the node number, and the vertical coordinate is the betweenness value. The value is CB v [0,0.1334] and the average value is 0.0011. It is calculated that only 2.2% stations with CB v 0.02 .
Fig. 4. Betweenness values of nodes of Xiamen bus network in double logarithmic scale.
(3) Clustering coefficient. Clustering coefficient is an important property of characterizing the local cohesiveness of the current node or the extent to which the nodes in the network are clustered together. It can be defined as C (vi )
2ei mi (mi 1)
(4)
where ei is the actual number of edges shared with local neighbors of node vi , mi is the connection degree of neighbors of node vi . Fig. 5 shows the values of clustering coefficient in Xiamen bus network. The horizontal coordinate is the node number. The values are distributed from 0 to 1 and the average value is 0.18. The stations with value C (vi ) 0.7 account for 6.3%.
Clustering coefficient
1 0.8 0.6 0.4 0.2 0 0
500
1000 1500 2000 2500 Node Fig. 5. Values of clustering coefficient of nodes sorted in Xiamen bus network.
4.1 Proposed indicators In networks, nodes have tendency to connect other nodes with similar type is called assortative networks, while nodes’ tendency to connect other nodes with different type is called disassortative networks [29]. The degree correlation plays an important role in traffic dynamics and self-organized criticality [30, 31]. Moreover, initial transfer times between nodes are also very important in a bus network. In this paper, we propose two indicators to evaluate the importance of nodes: neighborhood degree ratio and transfer index. (1) Neighborhood degree ratio. Neighborhood degree ratio i is defined as
i =
ki kj
Ni
where k i is the degree of the node i , N i is the set of neighbors of node i , and
(5)
kj
Ni
is the average degree of its neighbors. The large the value is, the important the
node is. Fig.6 exhibits the values of neighborhood degree ratio in Xiamen network ranging from 0.0175 to 2.9091. It is calculated that the average neighborhood degree ratio is 0.81. The stations with i 2 account for 1.85%. 3 2.5
i
2 1.5 1 0.5 0 0
500
1000 1500 Node
2000
2500
Fig. 6. Neighborhood degree ratios of nodes sorted in Xiamen bus network.
(2) Transfer index i . Node transfer index reflects the ability of accessibility to other nodes. It can be defined as N
i = ( j=1
1
)/ N stt ij 1
(6)
where sttij is the smallest transfer time from node i and node j . According to the definition, the element 0 i 1 . Large value means high ability of accessibility. Fig. 7 shows the values of node transfer index in Xiamen bus network sorted from 0.2828 to 0.6197. The average value is 0.42, and the stations with i 0.55 account for 2.6%。
0.65 0.6 0.55
i
0.5 0.45 0.4 0.35 0.3 0.25 0
500
1000 1500 Node
2000
2500
Fig. 7. Values of node transfer efficient sorted in Xiamen bus network.
4.2 Node failing process In this paper, we propose a node failing process to identify the hub nodes in bus networks. The process is called deliberate attack. When a node is attacked, all the edges that connected with the node under space P will be removed. Once the node has been attacked, the transfer condition of network will change. We use to represent the proportion of zero and one transfer time in a bus network, which can reflect the network transfer efficiency. There are six strategies to attack nodes in network deliberately according to degree, clustering coefficient, betweenness, neighborhood degree ratio, transfer index and random choice. There are two steps of the process. Take degree for example, (ⅰ) sorting the nodes with degree from large to small firstly, (ⅱ) attacking nodes step by step. 0.44
0.42
0.4
0.38
0.36
0.34
0.32 0
Transfer index random Neighborhood degree ratio Clustering coefficient Degree Betweenness 10
20
30
40
50
Step Fig. 8. Proportions of zero and one transfer time in Xiamen bus network with 50 different node
failing processes.
Fig. 8 illustrates the proportions of zero and one transfer ( ) in Xiamen bus network with 50 different node failing processes. The horizontal coordinate is the failing step, the vertical coordinate is the proportion of zero and one transfer time of the bus network. As can be seen, the decreases as the nodes been attacked. Take degree for example, are 0.4303 , 0.4193, 0.4112, 0.4020, 0.3883 and 0.3696 at step 0, 10, 20, 30, 40 and 50 respectively. Moreover, it is found that attacking nodes with large transfer index is more effective to destroy the network compared with other indicators. It is can be seen in the figure, are 0.3376, 0.3626, 0.3696, 0.3848, 0.4118 and 0.4133 after 50 steps for transfer index, neighborhood degree ratio, degree, betweenness, random and clustering coefficient respectively. In addition, the results also indicate that the bus network is robust by large clustering coefficient and random attack. To better understand the impact of attack on transfer time of the bus network, this paper investigates the proportions of zero, one, two, three, four and more transfer times after each attack. Fig. 9 shows the proportions by node failing process with large transfer index. It can be noted that the proportions of zero, one and two transfer times decrease as the nodes failing. For example, the proportions of two transfer times are 0.5170, 0.4999, 0.4944, 0.4931, 0.4902 and 0.4892 after failing 0, 10, 20, 30, 40 and 50 nodes respectively. While the proportions of three and more transfer time increase as the nodes failing. For example, the proportions of four times and more are 0.0051, 0.0161, 0.0264, 0.0357, 0.0472 and 0.0563 after failing 0, 10, 20, 30, 40 and 50 nodes respectively. 0.7 0.6
Proportion
0.5 0.4 0.3 0.2
Zero Transfer One transfer Two transfers Three transfers Four transfers and more
0.1 0 0
10
20
30
40
50
Step
Fig. 9. Proportion of zero, one, two, three, four and more transfer times in Xiamen bus network after node failing with large transfer index.
Table 1 shows the top 30 stations of Xiamen bus network with large transfer index. Station with large value means it is more important for network transfer efficiency. Failures of these stations can result in numerous unnecessary transfers, which enhance the total transfer time of the whole network. These stations are called hub stations in the bus network. Traffic managers should protect the traffic environment to enhance the efficiency of the bus network. Moreover, hub stations are also very useful in bus
network optimization. Table 1. Top 30 stations of Xiamen bus network with large transfer index. No.
Name
Value
No.
Name
Value
No.
Name
Value
1
SM Square
1340.3
11
Sibei Lukou
1249.0
21
Huoche Zhan
1231.3
2
Xingbei
1282.3
12
Zhaohu Zhan
1243.3
22
Haicang
1227.3
3
Banlian Guomao
1265.2
13
Chengxi
1241.0
23
Qiche Zhan
1225.3
4
Kaihe Lukou
1263.8
14
Dongdu
1240.5
24
Haiyu Lu
1224.3
5
Dianqian
1263.2
15
Lvcuo
1240.5
25
HC Jianhang
1224.3
6
Xiamen Bei
1258.8
16
Gaoqi Zhan
1240.2
26
Xilin
1224.0
7
Malong
1258.7
17
Shangjian
1239.0
27
Lianban
1222.7
8
Tangbian
1258.0
18
Lianhu Lukou
1234.8
28
Xinglin
1222.7
9
Xiaoxi Men
1253.7
19
Chengnan
1232.2
29
Huagong
1222.7
10
Neimao
1252.7
20
Xiangfeng
1231.7
30
Qinggong
1221.7
5. Identification of important lines Bus lines determine the transfer modes in a bus network. As the network become increasingly large, some lines play more and more important roles in transfers, which enhance the efficiency of bus networks. Line evaluation is crucial for bus network design. In this paper, we use transfer accessibility to measure the importance of lines in a bus network. We firstly define the line transfer matrix as follows: Tmm tij
i j tij ij i, j 1, 2, 0 i j
(7)
,m (8)
where Tmm is the line transfer matrix, m is the number of lines, ij is the number of transfer nodes between node i and node j . So, line transfer accessibility i of line i can be defined as m
i tij
(9)
j 1
Obviously, the large value of line transfer accessibility means less transfer time to other lines. Fig. 10 gives a simple example of a bus network with four lines. There are four transfer nodes in the network. Table 2 shows the line transfer matrix of the network. According to formula (9), a 2 , b 3 , c 2 , d 1 . The results indicate that line b is an important line from the perspective of transfers.
b a
c
d
Fig. 10. A simple example of a bus network with four lines. Table 2. The line transfer matrix of the simple example. a b c d
Line No.
Summation
a
0
1
1
0
2
b
1
0
1
1
3
c
1
1
0
0
2
d
0
1
0
0
1
Summation
2
3
2
1
0
Fig. 11 illustrates the line transfer accessibility for all lines. The horizontal coordinate is the line number. It is can be seen that the values of are from 2 to 1609. Moreover, there are only 3.23% of lines with 1000 , which contains lines ‘651’, ‘659’, ‘655’, ‘792’, ‘791’, ‘455’ et al. This means these lines with large play more important roles in network accessibility. 1800 1600 1400 1200
1000 800 600 400 200 0 0
100
200
300
400 Line
500
600
700
800
Fig. 11. Line transfer accessibility for all lines sorted.
6. Conclusions This paper constructed two complex network models by space L and space P. Space L is used to calculate the basic complex network indicators such as degree, clustering coefficient and betweenness. Space P is used to calculate transfer times between nodes. Neighborhood degree ratio and transfer index have been proposed to measure
the importance of stations. Moreover, a node failure process has been developed to identify hub stations in the bus network. Xiamen bus network is used to test the node failure model. The results show that the proposed transfer index is more effective to identify hub station in a bus network compared with other indicators. In addition, line transfer accessibility has been proposed to measure the importance of lines. The general idea of identification of hub stations and important lines has potential application in urban transit network design and management. In the future research, more cities’ bus network would be collected to verify the accuracy of proposed model. The model will consider the real transit demand combined with network structure. In addition, dynamic evaluation of urban transit network is another interesting research.
Acknowledgement The Project was supported by the Doctoral funds of Shandong Jianzhu University (XNBS1614), open fund for the key laboratory for traffic and transportation security of Jiangsu province (TTS2016-02) and Shandong Provincial Natural Science Foundation (ZR2013EEQ014).
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