Switch of critical percolation modes in dynamical city traffic

Transcription

1 Switch of critical percolation modes in dynamical city traffic Guanwen Zeng 1,2, Daqing Li 1,2,*, Liang Gao 3, Ziyou Gao 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 Traffic and Transportation, Beijing Jiaotong University, Beijing , China 4 Department of Physics, Bar-Ilan University, Ramat Gan , Israel Percolation transition is widely observed in networks ranging from biology to engineering. While much attention has been paid on network topologies, studies rarely focus on critical percolation phenomena driven by network dynamics. Using extensive real data, we study the critical percolation properties of the dynamics of city traffic. We find that two modes of different critical percolation behavior appear in the same network topology under different traffic dynamics. Our study suggests that the critical percolation of city traffic has similar critical characteristics as small-world networks during non-rush hours or days-off, while it switches to the behavior of two-dimensional lattice percolation during rush hours in working days. We show that this difference can be interpreted by the fact that the road velocity on highways during non-rush hours or days-off is relative high representing effective longrange connections like in small-world networks. Our results might be useful for understanding and mitigating traffic congestion. I. INTRODUCTION Critical phenomena of complex networks ranging from biology to engineering have attracted widespread attention. Previous studies on percolation [1,2], epidemic spreading [3-6], the Ising model [7-9] etc. have uncovered the critical behavior of different system processes. Studying their critical phenomena can help to develop efficient strategies to keep systems away from undesired system states [10]. Different systems can be characterized by a unified approach if they have the same critical exponents, a characterization called universality class [11-13]. It is found that the universality class in percolation is related to the network structural properties including dimension [14] and dependence [15]. While previous studies have mostly focused on effects of network topology [16], critical percolation behaviors of network dynamics have been rarely studied. As a typical dynamical flow network, city traffic is frequently observed to have transitions between free-flow and congestion states [17-20]. Accordingly, many models have been applied to analyze the dynamical properties of these systems at macro [21-25] or micro [26-32] level. A recent study [33] has demonstrated that the mesoscopic dynamical organization process of the global city traffic can be regarded as traffic percolation, where the giant component emerges at the critical percolation threshold. In traffic percolation, local road flows can form clusters of free flow, which will span over the whole network scale at the critical threshold. The critical threshold can be used as a characteristic efficiency measure of the global traffic in a city at a given time during the day. This traffic functional network is dynamic and evolves in real time, and the critical threshold changes dramatically during a working day. However, a systematic study on the critical behavior of dynamic percolation of city traffic is still missing. A fundamental question can be raised: whether the critical percolation properties of city traffic during different traffic periods, such as rush hours and non-rush hours, belong to the same or different universality classes? The answer of the above question can help us to better understand the formation and dissipation of traffic congestion. II. TRAFFIC PERCOLATION PROCCESS Our research is based on real-time traffic data of the road network in Beijing. The road segments and the intersections are regarded as links and nodes of the network, respectively. The road network includes over 52,000 links (road segments) and 27,000 nodes (intersections). The velocity dataset covers 29-day realtime velocity records of road segments in Beijing in 2015, including a representative holiday period in China, the National Day (from Oct. 1st to Oct. 7th). Velocities (km/h) are recorded by floating cars (mainly taxies), with resolution of one minute. 1 / 5

2 FIG. 1. (Color online) Percolation processes of city traffic. (a)-(d) Percolation processes at different instants: (a)(c) Oct. 1st (holiday), (b)(d) Oct. 15th (workday). As q increases, the giant component ( ) of traffic flows decreases and the second largest cluster ( ) shows a maximal value at criticality q c(t) as expected in percolation transition [1,2]. Note that the quantities are rescaled in the figure. (e)(f) A typical example of the traffic percolation breakdown into clusters (e) below and (f) above the critical threshold is shown. In real traffic network, roads can be classified into different states according to their velocity level. A road is considered as functional only if its velocity level can meet a given demand. We set a variable threshold q (0 q 1) to denote the velocity demand of drivers at a given time. Then we normalize the velocity v ij(t) of the traffic flow from site i to site j at time t, and get the relative velocity r ij(t). Specifically, we sort the velocity data measured during a day for each road segment in increasing order, and choose the 95 percentile as the standard maximal value v ij m. Thus, r ij(t) = v ij(t) / v ij m. Note that v ij(t) can be different from v ji(t) due to the direction of the traffic flow. For traffic state in each road segment, we compare its relative velocity r ij(t) with the given threshold q. If r ij(t) > q, the road segment meets the demand and is considered functional; otherwise, the road segment is dysfunctional and removed from the original road network. In this way, we can construct a functional network of the traffic dynamics from the original topology for any given q. Obviously, if q = 0, every road segment satisfies the demand and the functional traffic network is the same as the original (topological) road network. On the other hand, if q = 1, the functional network becomes completely fragmented as almost all the links are removed from the original network. Hence, we can study the process of how the global functional traffic flow disintegrates into local clusters of traffic flows as the demand increases. As q 2 / 5 increases from 0 to 1, we can observe that the functional traffic network becomes diluted, which can be regarded as traffic percolation [33]. In traffic percolation, there exists a critical threshold value of q at each time of a day, denoted by q c(t), where the giant component ( P ) of the dynamical network breaks down into fragmented local clusters and the second largest cluster (SG) reaches its maximum [1,2]. In FIG. 1 we demonstrate the percolation process where the giant component shows a phase transition with increasing q. At q c(t) the second largest cluster shows a maximum. We also demonstrate in FIG. 1(e) and 1(f), the breakdown of the giant component at criticality. Here we show that the traffic percolation threshold varies at different instants. As seen from FIG. 1, the value of q c(t) on a workday (Oct. 15th, 2015) is lower than that of a representative holiday (Oct. 1st, 2015) at a morning rushhour instant (8:00am). However, at noon the results of the two days seem very similar. According to the definition, q c(t) reflects the breakdown of the global dynamical traffic. If a car is traveling with relative velocity above q c(t), it FIG. 2. (Color online) Critical threshold of traffic percolation. (a) Values of q c(t) during a day for days-off ( ) and workdays ( ) respectively, with the resolution of 30 minutes. (b) Distribution of q c(t) on rush hours of daysoff ( ) and workdays ( ). Rush hours here mean 7:30~8:30am and 17:30~18:30pm. will be trapped in local isolated clusters [FIG. 1(f)]. Thus q c(t) indicates the maximal relative velocity that one can travel within a major part of the city (i.e. the giant component of the functional traffic network), which reflects the global organization efficiency of the city traffic at a given time. Next, we study how q c(t) evolves with time during the day. FIG. 2(a) shows that q c(t) is significantly different at rush hours between days-off (including holidays and weekends) and workdays. That is, q c(t) on working days is much smaller than that of days-off during rush hours, whereas they are similar in other time. This point can be also validated by the distribution of q c(t) during rush hours in both periods in FIG. 2(b), which shows that the distributions of q c(t) are well separated. With the above results, we can conclude that during rush hours the percolation efficiency of the city traffic on workdays is significantly lower than that on days-off.

3 III. CRITICAL EXPONENTS Here, we calculate the size distribution of finite clusters of free flows near the critical threshold for specific periods. FIG. 3(a) and 3(b) manifest in the workdays the variation of size distribution of finite clusters in the percolation during rush hours and non-rush hours, respectively. At criticality, it is found that the size distribution of finite clusters seems to follow a power law [1]: ns ~ s τ. (1) Here s is the cluster size, n s is the ratio between the number of s-size clusters and the total number of clusters, and τ is the corresponding critical percolation exponent. This can also be seen in day-off results in FIG. 3(c). Above the threshold (i.e. q c(t) + 0.1, q c(t) + 0.2), only small clusters in which nodes are connected with high-speed links exist. Thus, the size distribution seems to decay faster than a power law. As q approaches to the critical threshold q c(t), a large cluster (i.e. the giant component) appears, which represents the global traffic flow in the traffic network. When q further decreases below q c(t) (i.e. q c(t) 0.1, q c(t) 0.2), the scale of the giant component increases, since more nodes are linked to the giant component. Accordingly, the size of finite clusters decreases, leading to a more skewed tail in the size distribution. This behavior further supports the critical percolation hypothesis of traffic flows [1,2]. Focusing on the traffic percolation behavior at criticality, we find that the critical exponent of size distribution (denoted by τ ) during rush hours in workdays is in general smaller than that during non-rush hours, with the values of 2.07 and 2.33, respectively. However, this difference of critical behavior during different periods does not appear in days-off as seen in FIG. 3(c). Moreover, the critical exponent during non-rush hours in working days is almost the same as that during days-off. These results therefore indicate that there exist multiple modes of percolation critical behaviors for different periods. We also calculate the specific critical exponents of each period. The results cover a range of 29 days in 2015, as shown in FIG. 3(d). Note that Oct. 10th is a Saturday, but is still a workday according to the day-off compensation policy. It is interesting to note that these specific results of τ are generally within the range of the theoretical results of mean field (e.g. small-world or ER networks) and lattice percolation (e.g. two-dimensional regular lattice), with the value of 2.50 and 2.05, respectively [1,2]. During rush hours in working days, the critical exponents are much smaller compared to other periods, with the value closer to the limiting case of lattices; on the contrary, some values of τ in the National Holidays are approaching to the mean field limit. At other instants, results are between these two limiting cases. All these results suggest that the city traffic running at different modes between mean-field percolation and two-dimensional lattice percolation. Therefore, we may conclude that the value of τ in real data might represent different universality classes under FIG. 3. (Color online) Percolation critical exponents of cluster size distribution. (a)(b) Size distribution of traffic flow clusters near criticality during (a) rush hours and (b) non-rush hours in 17 workdays. Results include size distribution at q c(t) ( ), q c(t) 0.1 ( ), q c(t) 0.2 ( ), q c(t) ( ) and q c(t) ( ). (c) Critical size distribution of traffic flow clusters at criticality during rush hours ( ) and non-rush hours ( ) in 12 days-off. (d) Values of τ in specific periods of every day, including: rush hours in days-off ( ), rush hours in workdays ( ), non-rush hours in days-off ( ) and non-rush hours in workdays ( ). Rush hours here mean 7:30~8:30am and 17:30~18:30pm, while non-rush hours are 11:00am~13:00pm on every day. The theoretical results of mean field for small-world (τ = 2.50) and two-dimensional lattice percolation (τ = 2.05) are also marked as horizontal lines. different traffic modes. IV. EFFECTIVE LONG-RANGE CONNECTIONS A following question that can be naturally raised is why the critical properties behave differently at different times although the network topology is the same? For spatially embedded networks including the traffic network, the appearance of long-range connections [34-41] is found to alter the percolation exponents [14]. We suggest here that for transportation systems, the highways during non-rush hours or days-off can be regarded as effective long-range connections in the traffic network from a dynamical perspective. The highways normally connect distant places, and designed for higher velocities. For a driver, during non-rush hours and days-off, it is usually faster to reach a distant place by highways rather than by other roads. A plausible hypothesis for the different modes during the different periods is that during days-off or non-rush hours in working days, there exist more effective long-range connections which relax the spatial two-dimensional constraints of the original system. During these periods, the traffic system approaches the trait of small-world networks. However, with heavy traffic congestion during rush hours in working days, the highways become congested and are effectively removed from the network. 3 / 5

4 To demonstrate our assumption, we pick out the highways of the road network, and calculate their average speed during rush hours in workdays and days-off. Indeed, we find that the amount of highways with high velocities (larger than 70km/h) is significantly different during the two periods, as shown in FIG. 4(a) and 4(b). On days-off the fraction of high-speed highways is found larger in FIG. 4(c) than that on workdays. These highways connect to each other to form long-range connections between distinct places as seen in FIG. 4(a). Next, we test our hypothesis by exploring the influence of long-range connections in lattice percolation model. For that we apply a link-rewiring model of a two-dimensional small-world network [36] and study its critical properties. The model network is a lattice with a given fraction f of rewired long-range connections, consistent with the fact that the total number of links keeps constant in a city traffic network during different periods. At every step we randomly choose a node and disconnect one of its links; then we rewire it to a random node in the whole twodimensional network. This process continues until the fraction f of rewiring links meets the given value. After constructing the model network, we analyze its percolation properties. The results are shown in FIG. 4(d). It suggests that the value of τ increases with the increasing fraction of rewired links. We can see that the values of τ are basically changing from 2.05 to 2.50, which correspond to the universality classes of lattice percolation and mean field, respectively. Moreover, τ changes rapidly as the fraction f increases below 0.2. The larger fraction of links are rewired in the lattice, the more similar the network is to the ER network [37]. Therefore, our results support the hypothesis that during rush hours in workdays the dynamical percolation of city traffic has a stronger tendency towards two-dimensional lattice percolation since it is nearly under a zero-fraction of longrange connections. Systems with high fraction of effective long-range connections can exhibit pronounced dynamical effects, which will not occur in systems with short-range interactions only. The existence of long-range connections can result in quasi-stationary states with slow relaxation in stable and non-equilibrium steady states in driven dynamics [42,43]. As seen here [FIG. 4(d)] a small fraction of long-range connection can affect the critical exponents, resulting in changes of the system university class. In analogy, for example, studies have been conducted on the behavior of the Ising model in a smallworld network [44,45], and find that the introduction of disorder (reflected by rewiring probability p of network links) can result in changing the system universality class, from pure low-dimensional system for p = 0 to mean-field like region for p > 0. For city traffic systems, the introduction of long-range connections (fast highways) can remarkably shorten the effective distance between two nodes, which can improve the distant dynamical FIG. 4. (Color online) Effective long-range (high speed) connections. (a)(b) Highways with average speed larger than 70km/h (colored in green) on rush hours of (a) days-off and (b) workdays. The district shown is a part of the Beijing city. (c) Cumulative velocity distribution of highways in days-off (black) and workdays (red), in the traffic network. We only focus on velocities larger than 60 km/h. (d) Critical exponent τ as a function of the fraction of rewiring links for percolation in a small-world model. transmission [46] on flow networks. As in our hypothesis, by adjusting effective long-range connections one may change the system to the desired universality class of city traffic (like during non-rush hours or days-off). V. CONCLUSIONS City traffic is generally a spatial-temporal system, hence it is essential to focus not only on the structural static topology of a road network, but also the organization of dynamical traffic whose demand changes from time to time and place to place during the day. Here we study the percolation transition of city traffic dynamics using high resolution real-time traffic data. By analyzing the cluster size distribution of city traffic, we find that the critical exponent is close to the result of two-dimensional lattice percolation during rush hours in workdays while close to small-world during other periods. Our findings suggest that a key point affecting the critical exponent of traffic dynamics is the fraction of the effective long-range connections represented by the free highways. With the aid of traffic management methods, it may be possible to shift the system to the desired critical universality class by adjusting the amount of effective long-range connections. For example, if appropriate actions can keep the fraction of effective long-range connections above 0.1, we may therefore expect percolation critical exponents around 2.4, which represents a universality class of less fragmented global traffic in the city transportation system. In this sense, our study may be useful for comprehending and controlling the traffic congestion in the realization of the Smart City. 4 / 5

5 ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China ( ). 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. L.G. thanks support from the Fundamental Research Funds for the Central Universities (2015JBM058) and the National Natural Science Foundation of China ( , and ). * daqingl@buaa.edu.cn [1] A. Bunde and S. Havlin, Fractals and disordered systems (Springer, 1991). [2] D. Stauffer and A. Aharony, Introduction to percolation theory (CRC press, 1994). [3] R. M. Anderson, R. M. May, and B. Anderson, Infectious diseases of humans: dynamics and control (Oxford University Press, 1992). [4] R. Pastor-Satorras and A. Vespignani, Physical Review Letters 86, 3200 (2001). [5] Z. Toroczkai and H. Guclu, Physica A: Statistical Mechanics and its Applications 378, 68 (2007). [6] S. Meloni, A. Arenas, and Y. Moreno, Proceedings of the National Academy of Sciences 106, (2009). [7] L. Onsager, Physical Review 65, 117 (1944). [8] H. E. Stanley, D. Stauffer, J. Kertesz, and H. J. Herrmann, Physical Review Letters 59, 2326 (1987). [9] A. Aleksiejuk, J. A. Hołyst, and D. Stauffer, Physica A: Statistical Mechanics and its Applications 310, 260 (2002). [10] J. Gao, B. Barzel, and A.-L. Barabási, Nature 530, 307 (2016). [11] S. N. Dorogovtsev, A. V. Goltsev, and J. F. Mendes, Reviews of Modern Physics 80, 1275 (2008). [12] R. Cohen and S. Havlin, Complex networks: structure, stability and function (Cambridge University Press, 2010). [13] B. Barzel and A.-L. Barabási, Nature Physics 9, 673 (2013). [14] L. Daqing, K. Kosmidis, A. Bunde, and S. Havlin, Nature Physics 7, 481 (2011). [15] R. Parshani, S. V. Buldyrev, and S. Havlin, Proceedings of the National Academy of Sciences 108, 1007 (2011). [16] D. Garlaschelli, G. Caldarelli, and L. Pietronero, Nature 423, 165 (2003). [17] D. Chowdhury, L. Santen, and A. Schadschneider, Physics Reports 329, 199 (2000). [18] D. Helbing, Reviews of Modern Physics 73, 1067 (2001). [19] B. Kerner, The physics of traffic (Springer, New York, Berlin, 2004). [20] C. F. Daganzo, Transportation Research Part B: Methodological 41, 49 (2007). [21] M. Lighthill and G. Whitham, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (The Royal Society, 1955), pp [22] I. Prigogine and R. Herman, Kinetic theory of vehicular traffic (Elsevier, New York, 1971). [23] G. F. Newell, Transportation Research Part B: Methodological 27, 281 (1993). [24] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Physical Review E 51, 1035 (1995). [25] B. S. Kerner, Physical Review Letters 81, 3797 (1998). [26] K. Nagel and M. Schreckenberg, Journal de physique I 2, 2221 (1992). [27] D. Helbing and B. A. Huberman, Nature 396, 738 (1998). [28] M. Treiber, A. Hennecke, and D. Helbing, Physical Review E 62, 1805 (2000). [29] T. Tang, H. Huang, S. Wong, and R. Jiang, Acta Mechanica Sinica 24, 399 (2008). [30] T. Tang, H. Huang, S. Wong, and R. Jiang, Chinese Physics B 18, 975 (2009). [31] J. Long, Z. Gao, H. Ren, and A. Lian, Science in China Series F: Information Sciences 51, 948 (2008). [32] J. Long, Z. Gao, X. Zhao, A. Lian, and P. Orenstein, Networks and Spatial Economics 11, 43 (2011). [33] D. Li, B. Fu, Y. Wang, G. Lu, Y. Berezin, H. E. Stanley, and S. Havlin, Proceedings of the National Academy of Sciences 112, 669 (2015). [34] A. Weinrib, Physical Review B 29, 387 (1984). [35] S. Prakash, S. Havlin, M. Schwartz, and H. E. Stanley, Physical Review A 46, R1724 (1992). [36] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). [37] M. Barthélémy and L. A. N. Amaral, Physical Review Letters 82, 3180 (1999). [38] C. Moore and M. E. Newman, Physical Review E 61, 5678 (2000). [39] S. Redner, Nature 453, 47 (2008). [40] M. Barthélemy, Physics Reports 499, 1 (2011). [41] D. Li, G. Li, K. Kosmidis, H. Stanley, A. Bunde, and S. Havlin, EPL (Europhysics Letters) 93, (2011). [42] A. Campa, T. Dauxois, and S. Ruffo, Physics Reports 480, 57 (2009). [43] F. Bouchet, S. Gupta, and D. Mukamel, Physica A: Statistical Mechanics and its Applications 389, 4389 (2010). [44] A. Barrat and M. Weigt, The European Physical Journal B- Condensed Matter and Complex Systems 13, 547 (2000). [45] C. P. Herrero, Physical Review E 65, (2002). [46] R. Klages, G. Radons, and I. M. Sokolov, Anomalous transport: foundations and applications (John Wiley & Sons, 2008). 5 / 5