Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 694956 | 6 pages | https://doi.org/10.1155/2013/694956

The Analysis of the Properties of Bus Network Topology in Beijing Basing on Complex Networks

Academic Editor: Yi-Kuei Lin
Received01 Jan 2013
Accepted27 Feb 2013
Published26 Mar 2013

Abstract

The transport network structure plays a crucial role in transport dynamics. To better understand the property of the bus network in big city and reasonably configure the bus lines and transfers, this paper seeks to take the bus network of Beijing as an example and mainly use space L and space P to analyze the network topology properties. The approach is applied to all the bus lines in Beijing which includes 722 lines and 5421 bus station. In the first phase of the approach, space L is used. The results show that the bus network of Beijing is a scale-free network and the degree of more than 99 percent of nodes is lower than 10. The results also show that the network is an assortative network with 46 communities. In a second phase, space P is used to analyze the property of transfer. The results show that the average transfer time of Beijing bus network which is 1.88 and 99.8 percent of arbitrary two pair nodes is reachable within 4 transfers.

1. Introduction

Complex networks have been successfully used in many real complex systems since the researches of small-world networks and scale-free networks [1, 2]. Many real complex systems have been well studied which include the actor network [13], WWW [2, 4], protein networks [5], and power grid [1, 3]. During the last few years, transport networks, such as subway network [6], airport network [79], and street network [10], have been studied by the complex approach.

As an important part of urban transport systems and a trip mode to alleviate the traffic congestion, bus network has been studied by an increasingly large number of researchers. Sienkiewicz and Holyst studied the public transport in 22 Polish cities and found that the degree distribution of these network topologies followed a power law or an exponential function [11]. Xu et al. analyzed the three major cities of China, and they found that there is a linear behavior between strength and degree [12]. Soh et al. contribute a complex weighted network analysis of travel routes on the Singapore rail and bus transportation systems [13].

In this paper, we investigate the Beijing bus network (BBN) with 722 lines and 5421 nodes. The data can be achieved from the Internet (http://www.mapbar.com/search/). In the network, the nodes stand for bus stations and edges (links) are the bus line connecting them along the route. In order to analyze the static properties of BBN, we use the so-called space L and space P to represent the BBN. The space L is mainly used to analyze the properties of degree distribution, cluster, the average shortest path, degree correlation, and community structure. The space P is mainly used to analyze the average transfer time.

This paper is organized as follows. In Section 2, we give the description of the construction of the network. Section 3 analyzes the main properties of the BBN with space L. In Section 4, we analyze the average transfer time of the BBN and use space P to calculate the smallest transfer times between any nodes in the network. Discussion and conclusions are given in Section 5.

2. The Construction of the Network

For easy calculation, there are some assumptions followed.(1)The station name is the unique identification in the network. Do not account for the condition that some stations have identical names but different parking place.(2)The stations have slight difference between upstream line and downstream line. In this paper, the construction of the undirected network is based on the stations of the upstream lines.(3)This paper does not consider the number of links between two stations and the frequency of bus; that is, it does not consider the weight of the network.

The bus network is usually represented by space L, space P [11], space B [14], space C [15]. This paper will study and the Beijing bus network based on space L and space P. Space L consists of nodes which stand for the bus stations and a link between two nodes exists if they are consecutive stops on the route. Space P is a link formed between any two nodes of a line. Figure 1 gives a schematic representation of space L and space P.

3. The Main Properties of the Beijing Bus Network under Space L

Here we represent network as a graph , where is the set of nodes and is the set of edges (links). is described by the adjacency matrix . If there exists an edge between nodes and, ; otherwise . is the number of nodes in the network. Figure 2 gives the Beijing bus network topology graph with 5421 nodes and 16986 links.

3.1. Degree

The degree of node is defined as the number of nodes that connected with the node , which reflects the importance of the node . In this paper, it refers to the number of bus stations with direct bus connecting with the current bus station. After calculation, the largest degree of the BBN is 21, the smallest is 1, and the average degree of all nodes is 3.13 which means one station averagely connects 3-4 stations in Beijing bus network. Table 1 shows partly the bus stations of larger degree of BBN.


Serial numberBus stationdegree

1Sanyuanqiao21
2Liu Li Qiao Dong20
3Beijing Xi station19
4Liu Li Qiao Bei Li18
5Beijing Zhangdong16
6Madian Qiao Xi16
7Xi Dao Kou16
8Chong Wen Men Xi15
9Guang An Men Nei14
10Zuo Jia Zhuang14
11Qianmen14
12Dabei Yao Nan14
13Tianqiao13
14Ma Dian Qiao Nan13
15Bei Tai Ping Qiao Xi13
16Si Hui station13
17Deshengmen13
18Xi Bei Wang13
19Xiyuan13
20Xin Fa Di Qiao Bei13

Here, we studied the proportion of the stations with the degree from 1 to 21. Figure 3 shows the degree distribution of the stations in the BBN, and we found that it follows a shifted power law distribution, , is the degree, and the number of stations with degree 1 accounts for 4.48%. It also shows that the degree of 99% of all station is smaller than 10.

3.2. The Average Shortest Path

The average shortest path is the property to reflect the efficiency of information circulating on the network. It is defined as,   is the average minimum shortest number of steps between all pairs of nodes, and is the shortest path between node and node.

3.3. Cluster Coefficient

Cluster 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. In the BBN, clustering coefficient reflects the ease of the bus transport among the neighboring bus stations of the current one. It is defined as  , where is the cluster coefficient of node, is the actual number of links between the neighbor nodes of the current node, and is the degree of node . The cluster coefficient of the network is.

3.4. Efficiency

The efficiency is the property to characterize the capacity of traffic, and it can be calculated with the formula  , where is the shortest path between nodeand node [16].

3.5. Degree Correlation

Degree correlation reflects the relationship between the degrees of nodes. Nodes with high degree tending to be connected with nodes with high degree are called assortativity. In contrast, nodes with high degree which have the tendency to be connected with low degree are called disassortativity. It can be calculated with the formula [17] whereis correlation coefficient and andare the degrees of the nodes at the ends of theth links, with .

3.6. The Community Structure

It is usually found that there are many communities in one complex network; within the community, there are many links, but between the communities, there are fewer links [18]. Newman and Girvan [19] gave a measure called modularity. For a division with communities, then define a matrix whose component is the fraction of edges in the original network that connects nodes in community to those in community. The modularity is defined to be, where indicates the sum of all elements of . It can be achieved that. indicates the community structure is not stronger than it would be expected by random chance. The larger the modularity is, the stronger the community structure is.

Table 2 shows the main properties of BBN. We can see that the average shortest path is 20.03, which means people can reach destination by averagely taking 20 stops, and it is obvious that the BBN exhibits a small-world property. The cluster coefficient of BBN is 0.142, which means the BBN is a sparse network. Furthermore, the correlated coefficient is 0.185, which validates the result in the paper [m6] that when the number of nodes is larger than 500, the network is usually assortative. In addition, the community structure of BBN is so obvious and the modularity is 0.905 with 46 communities.


Network parametersValue

Number of nodes ( )5421
Number of links ( )16986
Number of lines ( )722
Average degree (Ak)3.13
Average shortest path ( )20.03
Cluster coefficient ( )0.142
Efficiency ( )0.066
Correlation coefficient ( )0.185
Number of communities ( )46
Modularity ( )0.905

4. The Transfer Property of Beijing Bus Network under Space P

The transfer capacity is an important index to evaluate the performance of a bus network, and travelers always expect that they can reach the destination through the least number of transfers. In this paper, the average minimum transfer time is used to evaluate the performance of the transfer capacity. Usually, travelers cannot reach the destination without transfer for a long distance trip, and the minimum transfer time between any two nodes is specific. The average minimum transfer time is the average among all pair nodes.

is used to represent the specific bus network, where is the line,is the bus station, and is a matrix,if linestops at station; otherwise.

Table 3 shows the specific network given in Figure 1. We can achieve the minimum transfer time using Table 3. For example, search. Because there is no directed line between the node and, it needs transfer. From Table 3, we can see that through, and through, so we can achieve; that is, travelers need transfer at from to. Search. We can get the path through two transfers.



1111110000000
0010001111000
0000100000111

Using the aforementioned method, we can get the minimum transfer time between any two nodes and calculate the average minimum transfer time. But it becomes very hard when the scale of network is becoming huge. In this paper, we use the space P to solve the problem. Firstly, we need to construct the network under space P, where the weight of the network is 1. Secondly, the Floyd algorithm is used to achieve the shortest path between any two nodes. The shortest path value is the minimum line number that needs to use and the transfer time is the needed line number minus 1. Figure 4 gives the illustration of the aforementioned example. Figure 4(a) shows can reach by using the two dotted lines. Figure 4(b) shows can reach by using the three dotted lines.

Here, we study the BBN. Table 4 shows the stations that have most lines. It is found that the station that owns most lines is San yuan qiao which has 47 lines.


Serial numberStationLine numbers

1Sanyuanqiao47
2Liu Li Qiao Bei Li40
3Beijing Xi Zhan39
4Zuo Jia Zhuang38
5Liu Li Qiao Dong34
6Liu Li Qiao Nan33
7Dongzhimen Wai31
8Gong Zhu Fen Nan31
9Bei Da Di30
10Bei Tai Ping Qiao Xi29
11Jing An Zhuang29
12Xiajia Hutong29
13Qing He29
14Xiyuan29
15Beijing Zhangdong28
16Xi Bei He28
17Xiju28
18Si Hui Zhan28
19Liangmaqiao28
20Mu xi yuan qiao dong27
21Yan Huang yishu Guan26
22Yuquanying Qiao Xi26
23Dongwu Yuan25
24Guang An Men Nei25
25Qianmen25
26Wanshou si25
27Mu Xi Yuan Qiao Xi25
28Kandan Qiao25

Figure 5 shows the distribution of the proportion of the stations and the amount of line number; the result shows that it follow the exponent distribution, where is the number of lines. It is found that most stations of BBN own less than 10 lines and only 9 stations own more than 30 lines.

In this paper, the transfer time of BBN is studied by using space P. From Table 5, we can see that the most pair nodes are reachable through one or two transfers, and 99.85 percent of the pair nodes is reachable within four transfers and the average minimum transfer time. Usually, the larger the is, the worse the performance of the bus network is. In general, cannot be more than 2; otherwise, we can consider that the performance of the bus network is bad and travelers’ trip is inconvenient. The transfer time of BBN is a little large and there is a room for improvement.


atr012344 moreunreachable

Proportion0.01680.2830.5530.1410.00470.000130.00137

5. Conclusion

In this paper, space L and space P are used to analyze the static properties of Beijing bus network. Space L is used to research the main topology properties of the Beijing bus network. The results show the Beijing bus network has small cluster coefficient, scale-free feature, and assortative correlation and the community structure is obvious. Moreover, we research the transfer property using space P. The result shows that the accessibility of the Beijing bus network is good and the average minimum transfer time is 1.88, which is a little large. A convenient bus network needs less transfers and high performance, and how to reduce transfer time and enhance the bus network dynamical performance is a valuable research.

Acknowledgment

This paper is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20120009110016).

References

  1. D. J. Watts and S. H. Strogatz, “Collective dynamics of “small-world” networks,” Nature, vol. 393, pp. 440–442, 1998. View at: Google Scholar
  2. A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999. View at: Publisher Site | Google Scholar | MathSciNet
  3. E. Ravasz and A. L. Barabási, “Hierarchical organization in complex networks,” Physical Review E, vol. 67, Article ID 026112, 7 pages, 2003. View at: Publisher Site | Google Scholar
  4. L. A. Adamic and B. A. Huverman, “Power-law distribution of the World Wide Web,” Science, vol. 287, no. 5461, p. 2115, 2000. View at: Publisher Site | Google Scholar
  5. S. Maslov and K. Sneppen, “Specificity and stability in topology of protein networks,” Science, vol. 296, no. 5569, pp. 910–913, 2002. View at: Publisher Site | Google Scholar
  6. V. Latora and M. Marchiori, “Is the Boston subway a small-world network?” Physica A, vol. 314, no. 1–4, pp. 109–113, 2002. View at: Publisher Site | Google Scholar
  7. A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, “The architecture of complex weighted networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 11, pp. 3747–3752, 2004. View at: Publisher Site | Google Scholar
  8. R. Guimerà, S. Mossa, A. Turtschi, and L. A. N. Amaral, “The worldwide air transportation network: anomalous centrality, community structure, and cities' global roles,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 22, pp. 7794–7799, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. T. Jia and B. Jiang, “Building and analyzing the US airport network based on en-route location information,” Physica A, vol. 391, no. 15, pp. 4031–4042, 2012. View at: Google Scholar
  10. B. Jiang, “A topological pattern of urban street networks: universality and peculiarity,” Physica A, vol. 384, no. 2, pp. 647–655, 2007. View at: Publisher Site | Google Scholar
  11. J. Sienkiewicz and J. A. Holyst, “Statistical analysis of 22 public transport networks in Poland,” Physical Review E, vol. 72, Article ID 046127, 11 pages, 2005. View at: Publisher Site | Google Scholar
  12. X. P. Xu, J. H. Hu, F. Liu, and L. S. Liu, “Scaling and correlations in three bus-transport networks of China,” Physica A, vol. 374, no. 1, pp. 441–448, 2007. View at: Publisher Site | Google Scholar
  13. H. Soh, S. Lim, T. Zhang et al., “Weighted complex network analysis of travel routes on the Singapore public transportation system,” Physica A, vol. 389, no. 24, pp. 5852–5863, 2010. View at: Publisher Site | Google Scholar
  14. K. A. Seaton and L. M. Hackett, “Stations, trains and small-world networks,” Physica A, vol. 339, no. 3-4, pp. 635–644, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  15. C. von Ferber, T. Holovatch, Y. Holovatch, and V. Palchykov, “Public transport networks: empirical analysis and modeling,” The European Physical Journal B, vol. 68, no. 2, pp. 261–275, 2009. View at: Publisher Site | Google Scholar
  16. V. Latora and M. Marchiori, “Efficient behavior of small-world networks,” Physical Review Letters, vol. 87, Article ID 198701, 4 pages, 2001. View at: Google Scholar
  17. M. E. J. Newman, “Assortative mixing in networks,” Physical Review E, vol. 89, Article ID 208701, 4 pages, 2002. View at: Publisher Site | Google Scholar
  18. M. E. J. Newman, “Detecting community structure in networks,” The European Physical Journal B, vol. 38, no. 2, pp. 321–330, 2004. View at: Publisher Site | Google Scholar
  19. M. E. J. Newman and M. Girvan, “Finding and evaluating community structure in networks,” Physical Review E, vol. 69, Article ID 026113, 15 pages, 2004. View at: Publisher Site | Google Scholar

Copyright © 2013 Hui Zhang 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.

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