Scale-free Resilience of Real Traffic Jams

Transcription

1 Scale-free Resilience of Real Traffic Jams Limiao Zhang 1, 2, Guanwen Zeng 1,2, Daqing Li 1, 2, *, Hai-Jun Huang 3, *, Shlomo Havlin 4 1 School of Reliability and Systems Engineering, Beihang University, Beijing , China 2 Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing , China 3 School of Economics and Management, Beihang University, Beijing , China 4 Department of Physics, Bar-Ilan University, Ramat Gan , Israel The concept of resilience can be realized in daily traffic, representing the ability of transportation system to adapt and recover from traffic jams. Although resilience is a critical property needed for understanding and managing the organization balance in traffic networks, a systematic study of resilience as well as its accepted definition in real urban traffic is still missing. Here we define a city traffic resilience as the size of the spatio-temporal clusters of congestion in real traffic, and find that the resilience follows a scale free distribution in both, city road networks and in highways, with different exponents, but similar exponents in different days and cities. The traffic resilience is also revealed to have a novel scaling relation between the cluster size of the spatio-temporal jam and its recovery duration, independent of microscopic details. Our findings provide insight towards universal properties of traffic resilience for better forecasting and mitigation of congestions. 1

2 Increasing traffic congestion is an inescapable problem due to enhanced urbanization and growing metropolitan cities all over the world, from Los Angeles to Tokyo and from Cairo to Beijing 1, leading to potential high economic and social losses. Under various internal or external perturbations, ranging from a local flow fluctuation to a broken-down traffic light and up to extreme weather conditions, a small jam can develop into a large-scale jam in a domino-like cascading process 2. Given the uncertainty of disruptive system failures, the concept of resilience describes the system ability to withstand possible perturbations and recover to an acceptable functional level. Since Holling s definition from ecology 3, the resilience framework has been developed and applied in many disciplines, ranging from engineering, economics to social science For example, system resilience across different domains usually depends on its absorptive capacity, adaptive capacity and restorative capacity 13. Accordingly, the possibility and recovery process in various critical infrastructures have attracted much attention recently Especially, a resilient transportation system in the future smart city era will improve significantly life quality, the development of economic society and reducing environmental pollutions 18. Transportation systems having network topology, as one of the critical infrastructures, serve as the backbone for national economics and stability. System resilience has been studied in different traffic systems including city road, metro system, freight 2

3 transportation and aviation network Different frameworks have been proposed to evaluate the resilience of infrastructures and communities. For example, Vugrin et. al 13 proposed a quantitative model for assessing the resilience of systems under disaster through the evaluation of system performance and economic cost of recovery. Chang and Shinozuka 27 introduced resilience measures that relate expected losses in future disasters to a community s seismic performance objectives. While most of resilience indicators is scenario-specific and limited to certain system performance, the network topology 28-30, although critical for resilience, has not been considered in resilience studies of critical infrastructures and other complex systems. It is proposed to define for earthquakes the measure of resilience as the change in system performance over time 31, which is the well-known resilience triangle theory. It measures the resilience loss of a community due to an earthquake using, t 1 RL= [100 Q(t)]dt. (1) t o Here Q(t) represents the service quality (ranged between 0% to 100%) of the community, which starts to decrease at t 0 and may returns to its normal state (100%) at t 1. Though this method is presented in the context of earthquakes, the concept has been applied for other types of disturbances 13, However, traffic system in a city has a network structure and its resilience evolves both, in space and time, while the above-mentioned studies focus mainly on the temporal evolution of dimensionless performance neglecting failures in space. 3

4 Composed of a very large number of strongly interacting subunits,transportation systems are usually running out of equilibrium states with unpredictable of cascading failures outcome 35. Due to the longstanding debate whether system resilience is intrinsic 36, it is critical yet unknown if such systems with numerous interacting subunits have universal resilient behavior that is independent of microscopic details. Here we propose a new traffic spatio-temporal resilience measure, incorporating both spatial and temporal features of system adaptation and absorption, to explore the universality of traffic resilience. With extensive real traffic data, we find novel scaling laws and scale free distributions for the traffic resilience and recovery times. Our definition and results demonstrate and support the existence of a universal behavior behind traffic congestions independent of microscopic details. These scaling laws hold for different sizes of traffic jammed clusters, which can help to predict system restoration behaviors and develop corresponding resilience management methods. Definition of spatio-temporal resilience Our study uses real traffic data of Beijing and Shenzhen, which are two megacities that suffer from most severe traffic jams world-wide and particularly in China. Complex road topology, large traffic flow and various perturbations as well as the availability of big data make these two megacities ideal for urban traffic research of resilience. A dynamical traffic network can be constructed based on road topology 4

5 information and high resolution of evolving traffic velocity data (see Data Description for detailed information). Specifically, the links in the jammed cluster at a given time represent congestion roads, while nodes in the jammed cluster are the intersections between these congested roads. Considering together the temporal evolution, as well as the spatial two-dimensional traffic network, we can regard the jam as a three-dimensional spatial-temporal network cluster. Accordingly, a three-dimensional (two of space and one of time) cluster can be constructed considering the same jam during its entire lifetime. The three-dimensional jammed cluster is demonstrated in Fig. 1A, where all red links in the shadow belong to the same jammed cluster. Note that the connected clusters here do not necessary mean that any roads within a connected cluster are spatially connected at a given time instant. When a jammed cluster splits into two or more sub-clusters at a certain instant, all links and nodes in the sub-clusters still belong to the same three-dimensional cluster due to their temporal connection. Our definition of jammed clusters intuitively grasps the spatio-temporal propagation and dissolution of traffic jams, instead of earlier dimensionless resilience indicators. We define the resilience based on the three-dimensional cluster size, using Eq. (2) in the following, for each jammed cluster during the observed period, e.g. from 6:00 to 22:00. For each jammed cluster, the number of its links (roads) at a snapshot of the temporal layer t, M s (t), varies with time. For the definition of jammed roads see 5

6 Data Description. Thus, M s (t) can be regarded as the cross section area of the jammed cluster at time t. Larger M s (t) means that more roads are congested at snapshot t. We can evaluate the resilience performance of the traffic network by analyzing the evolution and statistics of M s (t). For example, the time evolution of M s (t) of the second-largest jammed cluster on Oct. 26 th 2015 is demonstrated in Fig. 1B. The time span between t 0 and t 1, which is the lifetime of this jammed cluster, is defined as the recovery duration (T = t 1 t 0 + 1). The recovery duration reflects how long it takes for this jammed cluster to recover from the congestion. We define the cluster size, S, as the total number of links (roads) in the jammed cluster during its recovery time as, t 1 S = M s (t)dt. (2) t 0 The cluster sizes of the first three largest jammed clusters on Oct. 26th 2015 in Beijing as a function of time are shown in Fig. 1C. The cluster size naturally represents the loss of resilience in the traffic network. Eq. (2) does not only characterize the propagation of congestion in spatial dimension, but also represents the duration of congestion. Thus, the larger the jammed cluster size is, the less resilient the traffic system should be regarded. The shadow area shown in Fig. 1B represents therefore this loss of traffic resilience. To show the daily variations in the cluster sizes, we plot the size of the first three largest clusters as a function of date in Beijing (see Extended Data Fig. 1). The largest cluster sizes are found obviously 6

7 smaller in holidays (Oct. 1 st - Oct. 7 th ), due to the less traffic demand compared to normal workdays. Scale-free property of cluster size and recovery duration Next, we explore the distribution of cluster sizes and recovery duration in a typical day. The results on Monday Oct. 26 th 2015 in Beijing are shown in Fig. 2A. The distribution of cluster size shows a scale-free property, i.e., a power law scaling, P(S)~S α, (3) with an exponent α close to The power law distribution of cluster size suggests that although most of the congestions are of small-scales, there exist every day few congestions of sizes at all of the scales including extremely large spatio-temporal scale. With the signal-control based traffic management, small jams due to fluctuating traffic demand or accidents in a city will usually shrink and dissolve after a short timespan. However, if the traffic supply under real-time management cannot meet the increasing traffic demand, traffic jam will grow to a large scale and take more time to recover. These two behaviors compete in different scales in the city and possibly lead to the scale-free distribution of traffic resilience. Note that we obtain very similar results for Shenzhen, another megacity in China, during the same period (Fig. 2C). We also find that the cluster size distribution follows a very similar power law (α = 2.35 ± 0.02) for all observed workdays (see 7

8 Fig. 3). The high-quality scaling laws found here in different cities and different periods means that resilience defined here reflects an intrinsic property of urban traffic and is independent on the microscopic traffic details that change from day to day or even city to city. Since all sizes seem to follow the same scaling law, a unified mechanism may exist for different sizes and locations of jams. Next, we analyze and explore the scaling properties of the recovery duration in traffic congestion. In Fig. 2B, we show the distribution in a typical day in Beijing. It is found that the recovery duration of jammed clusters follows a distinct power-law distribution, Φ(T)~T β, (4) with an exponent β. Thus, the recovery duration also follows a power law distribution in all observed days with β = 3.10 ± 0.03 (see Fig. 3A). Similar results for scaling exponent, are also found for different cities and periods (Fig. 2D and Fig. 3B). Similar to the results for the cluster sizes, the power law distribution of recovery duration suggests that most of jammed clusters in the day recover relatively quickly. However, there exist all scales of recovery duration including some cases of very long recovery duration, where all (short, medium and long recovery durations) follow the same scaling law. This further indicates that the same general mechanism exists for all sizes of jams. Moreover, the scaling laws enable us to predict the recovery duration 8

9 for a given size of jammed cluster, which would be helpful for mitigation information guidance. Surprisingly, the power law exponents of resilience cluster size and recovery duration distributions are found stable in different days and different cities during the observed period (see Fig. 3). The appearance of the power law and its stability in different working days for a city is probably due to the self-organized nature 37 of traffic flow and corresponding management in urban traffic. On one hand, a large number of vehicles rush into the road network during peak hours, which leads to local congestions that propagate globally. Once the traffic flow returns to normal status, congestions disappear spontaneously. On the other hand, traffic control strategies such as traffic diversion, traffic lights and speed limitation are applied to alleviate traffic jams and pursue the system efficiency 38. All of these push the system towards the operational limits and probably contribute to our finding of robust scale-free distributions of cluster sizes and recovery duration. The universal mechanism suggested by the scale-free nature of traffic resilience usually depends on a few macroscopic variables including network dimension 39 and total traffic demand. To test this hypothesis, we also analyzed the traffic data of Beijing-Shenyang Highway between Oct. 1 st 2015 to Oct. 7 th The observed timespan is the National Day holiday in China, during which the highway is usually 9

10 under heavy traffic burden. The highway can be regarded as a one-dimensional road network, and the jammed clusters in the highway are therefore two-dimensional (one of space and one of time). As can be seen from Fig. 2E and Fig. 2F, the distributions of cluster size and recovery times of the two-dimensional jammed clusters also show a clear scale free scaling, but with different typical exponents. As seen in Extended Data Fig. 3, the scaling exponents are also surprisingly stable and almost do not change from day to day. For the one-dimensional highway, the scaling exponent for traffic resilience is much smaller than that of two-dimensional city, suggesting higher chance of larger jam. This lower resilience (large jams) is probably due to the fact that in jammed highways no alternative routing paths are available for cars, while jams in city traffic network have more opportunities to be resolved. On the other hand, as shown in Extended Data Fig. 2, resilience in holiday is higher with a higher exponent (2.69 ± 0.06) between Oct. 1 st and 7 th (the National Day of China) with significantly decreased total traffic demand. Scaling relationship between resilience and recovery duration In order to understand the relationship between traffic resilience and recovery duration, we show in Fig. 4A that the recovery time of jammed clusters increases with cluster size with a scaling relation, T~S γ, (5) 10

11 where γ is the scaling exponent. This scaling exponent is found similar between Beijing and Shenzhen. We also test the relation between cluster size and recovery duration of jammed clusters of the Beijing-Shenyang Highway (see Extended Data Fig. 4C) and find a different power law relation. The standard deviation in both cases is calculated from the γ values obtained for all observed days as shown in Extended Data Fig. 4. Next, we ask if these three exponents α, β and γ can be theoretically related. Indeed, if we assume P(S)~S α, Φ(T)~T β and T~S γ ( α, β, γ >0), the exponents α, β, γ should be related through the relation between the distributions 40, from which we obtain, P(S) = Φ(T) dt ds, (6) γ = α 1 β 1. (7) Indeed, Eq. (7) is valid within the error bars found for these exponents (see the comparison of actual value of γ with theoretical value, Eq. (7), of γ in Extended Data Fig. 4). Discussion In summary, we have developed a novel and intuitive definition of traffic resilience based on the spatio-temporal evolution of jammed clusters. We find, based on real data that both cluster size of jams and their recovery duration follow a scale-free 11

12 distribution suggesting universality principles of different congestion scenarios. This scaling relation is predictable and independent of traffic origin destination tables. The currently absence of common definition of traffic resilience could have been the reason of preventing efficient allocation of protection resources and policy design for mitigations. Our findings suggest that urban traffic systems could be organized into a few classes, with each class being described by the same scaling functions and the same set of scaling exponents. This result is of great theoretical interest, which motivates, in analogy to critical phenomena and the universality principle, theoretical studies regarding the intriguing question: Which traffic quantities are critical for determining the scaling exponents and scaling functions, and which are irrelevant? Moreover, the indications found here of universality in traffic resilience is also of great practical interest. Specifically, when designing resilience management methods 41-43, one may pick the most tractable traffic jam to study and obtain results for all other jams in the same universality class. The relationship between the cluster size and recovery duration can be applied to predict 44 the congestion influence and the duration of a certain jam size, which can help the decision-making in the future smart management of transportation. Meanwhile, further studies including simulations are needed to test and confirm the universal characteristics of our results. Future works should also focus, based on our novel approach, on evaluating the traffic resilience in other cities and other infrastructures when appropriate data becomes available. With 12

13 the broad range of applications of urban traffic analysis, developing innovative interdisciplinary approaches based on big data to identify and understand the origin of the scaling laws 45,46 of system resilience is thus a future big challenge. 13

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19 Acknowledgements We thank the support from the National Basic Research Program of China (2012CB725401). D.L. acknowledges support from the National Natural Science Foundation of China ( ) and the Fundamental Research Funds for the Central Universities. S.H. thanks the Israel-Italian collaborative project NECST, Israel-Japan Science Foundation, the Israel Science Foundation, ONR, DTRA (Grant No. HDTRA ) and BSF-NSF for financial support. We thank the data provider Beijing PalmGo Infotech Co., Ltd. Author Contributions DL and LZ conceived and designed the research. LZ, DL, GZ acquired and analyzed the data. HH, DL and SH interpreted the results. All authors discussed, wrote, and approved the final version of the manuscript. Author Information Correspondence and requests for materials should be addressed to D.L. or H.H. Data Availability 19

20 Due to privacy issues, we agree not to spread data publically according to the limitations of collection licensees. Meanwhile, our data are available from the corresponding author on reasonable request. Data Description We describe here in detail the construction of the dynamical traffic networks based on road topology information and high resolution of evolving traffic velocity data. The network in Beijing contains over 39,000 road segments (links) and 27,000 intersections (nodes), while Shenzhen traffic network contains about 18,000 road segments (links) and 12,000 intersections (nodes). The data set covers velocity records in both cities for 30 days during October 2015 with resolution of one minute, which are recorded through floating cars. Each road in the traffic network has a velocity v i (km/h) and a given velocity threshold p i is determined to judge the traffic status on this road (detailed thresholds for different roads are shown in Extended Data Table 1). Then roads with real-time velocity v i below threshold are regarded as congested. 20

21 Fig. 1 Traffic resilience defined from jammed clusters. A. Illustration of the evolution of a jammed cluster. Green links are in functional state, while red links are considered congested. All red links in the shadow belong to the same jammed cluster. B. The cross section area Ms(t) of the second-largest jammed cluster on Oct. 26th 2015 in Beijing. Since the resilience is reduced during the jam, we plot the negative of Ms(t) as a function of time, and traffic resilience can be represented by the grey area. The grey area is the size of the spatio-temporal jammed cluster (S), shown in red in A. Time span between t 0 and t 1 represents its recovery time (T = t 1 t 0 + 1). C. The cluster sizes of the first, second and third-largest jammed cluster on Oct. 26th 2015 in 21

22 Beijing as a function of time (the second and third largest clusters sizes are given on the right axis scale). 22

23 Fig. 2 Scale free distributions of traffic resilience: A. The distribution of jammed cluster size. B. The distribution of recovery duration. A and B are typical results based on city traffic data in Beijing on Oct. 26 th C and D are typical results based on city traffic data in Shenzhen on Oct. 26 th E and F are typical results based on traffic data of Beijing-Shenyang Highway on Oct. 1 st The results are analyzed by logarithmic bins, and plotted in double-logarithmic axis. 23

24 Fig. 3 Scaling exponents of the scale-free distributions of cluster size and recovery time as a function of date A. in Beijing, B. in Shenzhen. 24

25 Fig. 4 Typical scatter plots of recovery time versus cluster size A. in Beijing on Oct. 26 th 2015 and B. in Shenzhen on Oct. 26 th