Time series analysis with mixed traffic flow : A complex visibility graph method
Shangyi Xua, Huijun Suna, Jianjun Wub
a
MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, PRC
E-mail: 08121234@bjtu.edu.cn
b State Key Laboratory of Rail Traffic Control and Safety , Beijing JiaotongUniversity, Beijing 100044, PRC
E-mail: sunhuijun@jtys. bjtu. edu.cn
Contributions
Results
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).
Method
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:
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.
of different vehicle densities S=0.05, 0.1, 0.5, when
R=0.01.
Vehicle
densities
L
C
0.05
0.1
0.15
0.2
0.25
0.3
0.35
a
41.901 10.641 7.506 7.004 7.011 7.011 6.657
0.383 0.361 0.354 0.382 0.401 0.406 0.409
Vehicle 0.05
0.1
0.15 0.2
0.25
0.3
0.35
densities
L
49.490 10.611 7.271 7.023 6.664 6.665 6.644
Tab.1 The average shortest path and clustering coefficient of the
evolving networks for different initial vehicle densities, when
R=0.01 (R is the ratio of slow vehicles).
Vehicle 0.05
0.1
0.15 0.2
0.25 0.3
0.35
densities
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 The average shortest path and clustering coefficient of the
evolving networks for different initial vehicle densities, when
R=0.06.
References
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[2] Watts D.J. and Strogatz S.H., Nature (London), 393
(1998) 440.
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(2004) 147.
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[6] Gong P. and van Leeuwen C., Physica A, 321 (2003)
679.
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Nuńo C., PNAS, 105 (2008) 4972.
of different vehicle densities S=0.05, 0.1, 0.5, when
R=0.01.
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
(x , y )
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.
a
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.
Fig.2 The degree distributions and evolving networks
Fig.1 The degree distributions and evolving networks
C
0.386 0.351 0.365 0.382 0.399 0.404 0.418
Tab.3 The average shortest path and clustering coefficient of
Fig.3 Lane-change probability pn.change1=1
and pn.change2=0.5 between two lanes , when
R=0.01.
the evolving networks for different initial vehicle densities and
lane-change probability pn.change1=1 and pn.change2=0.5 between
two lanes when R=0.01.
Conclusions
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.
Funding
This paper is partly supported by National Basic Research Program of China (2006CB705500),
NSFC of China (70801005 and 70871099) and FANEDD (200763).