Study on Network Characteristics of Cross-Border Traffic Network of Urban—A Case Study in Lanzhou Yong-sheng Qian School of Geography and Environmental Science Northwest Normal University School of Traffic and Transport Lanzhou Jiaotong University Lanzhou, China Email:
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Chun-fang Liu School of Geography and Environmental Science Northwest Normal University Gansu Research Academy of Land Resource Plan Lanzhou, China Email:
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Jun-wei Zeng School of Traffic and Transport Lanzhou Jiaotong University Lanzhou, China Email:
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Shou-bao Wang School of Traffic and Transport Lanzhou Jiaotong University Lanzhou, China Email:
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Abstract—In case that the traffic resource is limited, cross-border traffic has great impact on urban transportation, and it is necessary for us to study characteristics of Lanzhou cross-border traffic network. In this paper, the characteristics of cross-border traffic network are studied from different aspects based on the theory of complex network. Results show that Lanzhou cross-border traffic network has small word and scale-free characteristics. These results may bring useful advices for the planning of Lanzhou traffic network.
subway, railway, grid [8] and shipping networks, etc. Newman and Moore, May and Lloyd did jobs about virus transmission in the small world network and Internet network and dynamic virus transmission model of scale-free network [9-11]. Amaral L A N, Scala A, Barthelemy M, et al (2002) studied aviation network topology [12]. Latora and Marchiori (2002) have made a preliminary study on characteristics of the Boston subway network [13]. Sen, et al (2002) studied small-word characteristics of the Indian railway network [14]. Miao Wei, et al (2007) made a preliminary study on the complexity of the world shipping network [15]. GAO Zi-you, et al (2005) have made preliminary Study on the Complexity of Traffic Networks and Related Problems on urban transport networks, taking transit networks in Beijing for example [16-17] and (2006) pointed out a number of studying theme of the complexity of urban transport network [18]. The characteristics of cross-border traffic network are not studied both at home and abroad, judging from the current information.
Keywords-complex network; small-word scale-free network; cross-border traffic; valley-city
I.
network;
INTRODUCTION
The study of Complex networks achieves the eye-catching accomplishment in theoretical analysis, modeling and demonstration aspects. In the real world there are a lot of complex systems described by complex network, in which the nodes are defined as the individual of the system and the edges are defined as the interrelationship between individuals. In the 1950s, Hungarian mathematician Erdos and Renyi simulated realistic network using random network model [1] and simulation results show that the most realistic network evolution is gradually formed [2-5], but random network model can not explain these evolutionary mechanisms. 1998 Watts and Strogatz put forward a small world network model [6] and 1999 Barabasi, Albert and Jeong advanced scale-free network model [7]. That complex network is key factor of depicting and studying the topological structure and behavior of complex system has become one of scientist’s research hotspots. The characteristics of many realistic networks have been studied by many scholars at home and abroad, such as the Internet network, the virus transmission model, aviation,
Along with the rapid development of the national economy, the economic exchanges between cities become more closely and the volume of cross-border traffic rapidly increases. Lanzhou is a key transport hinge, so the cross-border traffic network plays an important role in the economic development. Along with the increase of the volume of traffic, cross-border traffic network’s traffic load has increased. Because Lanzhou is the valley-city being restricted by the special topographical conditions, the cross-border traffic and urban traffic are not separated, which causes traffic jam situation of cross-border traffic network. In case that traffic resource is limited, the analysis of characteristics of cross-border traffic network is inevitable path of improving the utilization efficiency and tension of urban traffic.
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ki = ¦ δ ij
In this paper, a case study on cross-border traffic network of Lanzhou from different aspects shows the characteristics of cross-border traffic network. The analysis of cross-border traffic network has an extremely important theoretical practical significance for preventing cross-border traffic network being undermined, controlling traffic congestion and improving systemic reliability of network. II.
So, the average degree of the network is defined as
k=
CHARACTERISTICS OF CROSS-BORDER TRAFFIC NETWORK
Lanzhou is one of the 45 highway traffic hinge cities, as well as it is 212, 213, 312, 109, 309 State Roads’ starting and cross-border point, cross-border transport network of which is shown on Fig. 1 [19]. Yellow River goes through Lanzhou, and both sides of Lanzhou is hill. Along with the vigorous development of western region in recent years, the cross-border traffic has a great impact on urban traffic in Lanzhou, so the study on cross-border traffic network of Lanzhou based on complex network theory is extremely important practical significance.
(3)
j∈G
1 N
¦k
i
=
i∈N
2E N
(4)
2˅ Average path length as
There is a certain path between any two nodes denoted i, j in the network, the shortest of which is called the
shortest path ( d ij ) between nodes ( i,
j ). The average
path length of a network is defined as the average value of the shortest path length between all the nodes ( i, j ), the formula is defined as
l=
2 ¦ dij N ( N − 1) i , j
(5)
i≠ j
3˅ Clustering coefficient Clustering coefficient is used to indicate the extent of the network compactness. For arbitrary node in network and the set of its adjacent nodes ( N i ), so the number of edges between arbitrary node and its next neighbors in set N i is defined as
Mi =
A. Static Statistics of Cross-border Transport Network For a given graph or network, defining the set of all nodes as N , defining the set of all edges as E, so the network is defined as G(N, E) Where
Ci =
2M i (ki − 1)ki
(7)
Therefore, the average clustering coefficient of network is defined as
(1)
Where
(6)
i is defined as
Clustering coefficient of node
Figure 1. Main cross-border routes of Lanzhou
N = {1,2," ,i," ,n} ˈE = {δ ij }
1 ¦ δ ijδ jlδ li 2 j ,l
C䰉
1 N
¦C
i
(8)
i∈N
4˅ Betweenness
1 ¯0
δij=®
(2)
Where 1 implies that there is a director route which connects i and j , 0 means that there is a direct route between i and j .
Definition of network topology parameter 1˅ Nodes degree Usually, the nodes degree is defined as the number of its next neighbors, so the formula is defined as
The betweenness of network includes node betweenness and edge betweenness. The node betweenness is defined as the number of the shortest path which passes through this node. The edge betweenness is defined as the number of the shortest path which passes through this edge. The number of node betweenness reflects influence of the node in the network, and the edge betweenness reflects the importance of edge in the network. The set of the shortest path between almost two nodes ( i, j ) is N ij , and the node ( m ) betweenness is defined as
Bm = ¦
¦
i, j
l∈ N ij
δ lm
(9)
N ij
The edge ( e ) betweenness is defined as
Be = ¦ i, j
¦
l∈Nij
δ le
N ij
(10) Figure 3. Degree distribution of node
B. Establishing Cross-border Traffic Network Model Cross-border traffic network is composed of routes and intersections (Fig. 2). A route goes through several intersections, but there are also a number of routes going through an intersection.
Figure 4. Degree distribution curve of node
Figure 2. Cross-border traffic network model
Cross-border traffic network model describes 32 main intersections and connecting routes in cross-border traffic network of Lanzhou. Because Lanzhou is the valley-city, cross-border traffic and urban traffic are sharing part of cross-border traffic routes. Analysis of the characteristics of cross-border traffic network is important to solve the problem of urban traffic caused by cross-border traffic. Therefore, it is necessary to make an analysis of the topological characteristics of cross-border traffic network combined with the characteristics of valley city. C. Analysis of Characteristics of Cross-border Traffic Network In this paper, cross-border traffic network of Lanzhou is analyzed as a no-weight network, the analysis and statistics as follows: The nodes degree distribution of cross-border traffic network in Lanzhou is shown on Fig. 3. That the nodes degree distribution of cross-border traffic network fits power law is the characteristic of scale-free network and different from the other networks in which the nodes degree distribution fits Poisson distribution. In this paper, we use mathematical software Matlab to establish fitted curve.
According to probability distribution of nodes P (k ) = ck − r , logarithmic transform of it is log (P ( k ) ) = log c + (− r )log k , obtaining c = 138.5, r = 4.75, so probability distribution of nodes degree is P (k ) = 138 . 5 k − 4 . 75 , being shown on Fig. 4. Because Lanzhou city is a valley city, many roads are built along river and the hill foot, so the number of nodes degree 3 is very large. According to the calculation of all-pairs shortest-paths, we get 1674 the shortest path. The longest shortest-paths length is 8 and the average shortest-paths length () is 3.740. The path length distribution is almost fitting Poisson distribution, being shown on Fig. 5. Being seen from the figure, the step between arbitrary two nodes is ranging from 3 to 4, almost fitting description of small-word network. The clustering coefficient of arbitrary nodes i in the network Ci (Fig. 6) and cross-border traffic network in Lanzhou C= 0.088 are obtained through computer simulation. That the majority of network nodes’ clustering coefficient is zero shows that nodes weakly link its next neighbors and the group extent of network is very weak, and the nodes of part of cross-border traffic network in the internal city have clustering coefficient, because the cross-border traffic network which is urban external main network is different from urban traffic network.
Figure 5. The shortest path length distribution
the small-word network and scale-free network. The study on characteristics gives a theoretical support to plan a reasonable cross-border traffic network. The results will benefit the improvement of cross-border traffic in order to reduce its impact on urban traffic and the utilizing efficiency of cross-border traffic network. ACKNOWLEDGMENTS The authors wish to thank the reference for his/her helpful comments and suggestions for improvement. Figure 6.
Clustering coefficient of node
According to calculation, in the Lanzhou, the node betweenness distribution is shown on the Fig. 7. Being seen from the Fig. 7, the largest of node betweenness is 734, and the average node betweenness is 280, showing that several nodes with large betweenness play an important role in network. For example, Yan Tan, Xiao Xi Hu and Bai Yin intersections bear a lot of cross-border traffic. According to the statistics of edge betweenness, the largest edge betweenness is 404 and the average edge betweenness is 166 in the network, but the smallest edge betweenness is 14. Therefore, the traffic load rate of road in the network is uneven. For example, Bin He Road, Bai Yin Road, Xi Jin Road and Zhong Shan Road bear a lot of cross-border traffic.
This paper is partly supported by the Science and Technology Project of Gansu province (096RJZA088). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12]
Figure 7. Node betweenness distribution
III. CONCLUSIONS Based on complex network theory, the characteristics of cross-border traffic network in Lanzhou are analyzed. Simulation results show that cross-border traffic network has some characteristics of small-word network and scale-free network, including Poisson distribution of average path length and power law distribution of node degree. The clustering coefficient of network is very small and most of node clustering coefficient is zero, so it has some difference between the ideal scale-free network and cross-border traffic network, but also confining the actual characteristics of cross-border traffic network, because that cross-border traffic network is affected by limited geographical conditions, transport planning and the other factors. Summing up, the analysis of cross-border network of Lanzhou City allows us to give a clear physical meaning of
[13] [14] [15]
[16]
[17]
[18]
[19]
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