Time series analysis with mixed traffic flow : A complex visibility graph method Time series analysis with mixed traffic flow : A complex visibility graph method Shangyi XuShangyi Xua, Huijun Sun, Huijun Sunaa, Jianjun Wu, Jianjun Wubb

a MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, PRCMOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, PRCE-mail: 08121234@bjtu.edu.cnE-mail: 08121234@bjtu.edu.cn

bb State Key Laboratory of Rail Traffic Control and Safety , Beijing JiaotongUniversity, Beijing 100044, PRC State Key Laboratory of Rail Traffic Control and Safety , Beijing JiaotongUniversity, Beijing 100044, PRCE-mail: sunhuijun@jtys. bjtu. edu.cnE-mail: sunhuijun@jtys. bjtu. edu.cn

ContributionsContributions

We introduce the visibility algorithm method, which can reproduce the complexity of time series. By mapping to the complex network from nonlinear traffic flow series, more clear physics meaning is given such as the degree distribution, the clustering coefficient and the average shortest path. The traffic dynamics can then be studied by investigating the statistical properties of complex networks topologies. When the vehicle density S=0.05, the constructed complex network displays the characteristics of the regular graph. We also found that the networks constructed by the visibility algorithm can exhibit characteristics of the small-world for the higher density (time series generated by the STCA model).

The simulations have been carried out with different ratio of vehicles, which are initially positioned randomly. The fast cars as well as slow cars may use both lanes, so that both cars are treated equally with respect to the lane-changing behavior. To study the correlated characteristics of complex network in different vehicle densities, we give the topologies and degree distributions of networks constructed by our model for different vehicle density in the Fig.1. This diagram represents the network properties of the traffic flow as a function of density. Obviously, it can be concluded from Fig.1 that as vehicle density gradually increase, the graphs and degree distributions seem little change. In addition, according Tab.1 we can see that with the increase of vehicle density, the average shortest path of the complex networks general decreases. At the same time, the clustering coefficient will be large with the increase of vehicle density. Additionally, from the Tab.1, when the vehicle density S=0.05 , we find the average shortest path L=41.901 and the clustering coefficient C=0.383. Obviously, the network does not exhibit the random and small-world network characteristics. But the regular graphs characteristics can be found. The potential of the new method introduced in this article is illustrated with respected to our correlation evolution mechanism for traffic flow time series and complex networks. Another interesting result is that as the increase of vehicle density, the network with the small-world property will be displayed.

ResultsResults

[1] Barabási A.L. and Albert R., Science, 286 (1999) 506.[2] Watts D.J. and Strogatz S.H., Nature (London), 393 (1998) 440.[3] Bollobás B., London, Academic Press, 1985.[4] Amaral L.A.N. and Ottino J.M., Eur. Phys. J. B, 38 (2004) 147.[5] Zhang J. and Small M., Phys. Rev. Lett., 96 (2006) 238701.[6] Gong P. and van Leeuwen C., Physica A, 321 (2003) 679. [7] Lacasa L., Luque B., Ballesteros F., Luque J. and Nuńo C., PNAS, 105 (2008) 4972.

ReferencesReferences

In this article we introduced a tool to analyze the complexity of time series named the visibility graph. This algorithm offers an interested method to map a network from a time series. As will be show below, this network inherits several properties of the time series, and its study reveals nontrivial information about the series itself, so that we understand the complexity of the traffic flow. In the algorithm, every node corresponds, in the same order, to series data, and two nodes are connected if visibility exists between the corresponding data, that is to say, if there is a straight line that connects the series data, provided that this “visibility line” does not intersect any intermediate data height . A graph is composed by nodes and edges. In the visibility criteria, two arbitrary nodes with data values (ta, ya ) and (tb, yb ) will have connectivity, if any other data placed (tc, yc ) between them fulfills:

We can easily check that by means of the present algorithm, the associated graph extracted from a time series is always : (1) Connected: each node sees at least its nearest neighbors (left and right). (2) Undirected: the way the algorithm is built up, there is no direction defined in the links. (3) Invariant under affine transformations of the series data: the visibility criterion is invariant under rescaling of both horizontal and vertical axes, and under horizontal and vertical translations.

Method Method

Fig.1Fig.1 The degree distributions and The degree distributions and evolving networks of different vehicle evolving networks of different vehicle densities S=0.05, 0.1, 0.5, when R=0.01. densities S=0.05, 0.1, 0.5, when R=0.01.

Fig.3 Fig.3 Lane-change probability Lane-change probability ppn.change1n.change1=1 and =1 and ppn.change2n.change2=0.5=0.5 between between

two lanes , when R=0.01two lanes , when R=0.01. .

Fig.2Fig.2 The degree distributions and The degree distributions and evolving networks of different vehicle evolving networks of different vehicle densities S=0.05, 0.1, 0.5, when R=0.01. densities S=0.05, 0.1, 0.5, when R=0.01.

In conclusion, consideration the recently addressed problems of “topology evolution” and “traffic flow model” in complex networks, we introduce the visibility algorithm to construct the network from the traffic flow time series generated by STCA model. It shows that the constructed networks exhibit the small-world characteristics effects with the increase of traffic density. But for the vehicle density S=0.05 , the network will display the properties of the regular graph. Additionally, some of important statistical characteristics of the constructed networks are discussed for different lane-change probabilities. It is found that different lane-change probabilities almost have no great influence to the complex network properties.

ConclusionsConclusions

Tab.1Tab.1 The average shortest path and clustering The average shortest path and clustering coefficient of the evolving networks for different coefficient of the evolving networks for different initial vehicle densities, when R=0.01 (R is the ratio initial vehicle densities, when R=0.01 (R is the ratio of slow vehicles).of slow vehicles).

The next step is to analyze the effects of the ratio of slow vehicles on the characteristics of complex networks. According Fig.2 and Tab.2, we find that the average shortest path L=84.195 and the clustering coefficient C=0.395, when the vehicle density S=0.05 and the ratio of slow vehicle R=0.06. Therefore, the network exhibits characteristics of the regular network. In addition, we also find that, when the vehicle density S=0.1, the same result can be draw as S=0.05. From Tab.2, when the vehicle density S=0.15 and the ratio of slow vehicle R=0.06, the network exhibits the properties of the small-world. At the same time, with increasing density the networks show the small-world properties more oblivious. Therefore, the results demonstrate that the different ratio of slow vehicle have little effect on the visibility method in mapping time series into complex network topologies. It can be concluded from Fig.3 and Tab.3 that the characteristics of the graphs have no obviously changes under the symmetry and asymmetry lane-change conditions compared with Fig.1 and Tab.1. In addition, according the Tab.3, when the vehicle density S=0.05, we find the average shortest path L=49.490 and the clustering coefficient C=0.386, which indicates the network with the regular graphs characteristics. But when the vehicle density S=0.25, the average shortest path L=6.664 and clustering coefficient C=0.399 are obtained, which shows the network with the small-world characteristic. Moreover, the higher the vehicle density S, the more obvious of the small-world effects will be revealed. In a word, from Fig.3 and Tab.3 obtain that different lane-change probabilities have little influences on the visibility algorithm in mapping time series into complex networks.

Tab.3 The average shortest path and clustering The average shortest path and clustering coefficient of the evolving networks for different coefficient of the evolving networks for different initial vehicle densities and lane-change initial vehicle densities and lane-change probability probability ppn.change1n.change1=1 and =1 and ppn.change2n.change2=0.5=0.5 between between two lanes when R=0.01.two lanes when R=0.01.

This paper is partly supported by National Basic Research Program of China (2006CB705500), NSFC of China (70801005 and 70871099) and FANEDD (200763).

FundingFunding

( , )a ax y

Vehicle densities

0.05 0.1 0.15 0.2 0.25 0.3 0.35

L 41.901 10.641 7.506 7.004 7.011 7.011 6.657C 0.383 0.361 0.354 0.382 0.401 0.406 0.409

Vehicle densities

0.05 0.1 0.15 0.2 0.25 0.3 0.35

L 84.195 25.883 6.824 6.659 6.897 6.800 6.866

C 0.395 0.382 0.375 0.380 0.389 0.393 0.395

Tab.2 Tab.2 The average shortest path and clustering The average shortest path and clustering coefficient of the evolving networks for different coefficient of the evolving networks for different initial vehicle densities, when R=0.06.initial vehicle densities, when R=0.06.

Vehicle densities

0.05 0.1 0.15 0.2 0.25 0.3 0.35

L 49.490 10.611 7.271 7.023 6.664 6.665 6.644

C 0.386 0.351 0.365 0.382 0.399 0.404 0.418