Percolation properties in a traffic model
FEILONG WANG1,2, DAQING LI1,2 (a), XIAOYUN XU1,2, RUOQIAN WU1,2 and SHLOMO HAVLIN3
1 School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China 2 Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China 3 Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
PACS 89.75.-k – Complex systems PACS 89.75.Fb –Structures and organization in complex systems PACS 05.10.-a – Computational methods in statistical physics and nonlinear dynamics Abstract – As a dynamical complex system, traffic is characterized by a transition from free flow to congestions, which is mostly studied in highways. However, despite its importance in developing congestion mitigation strategies, the understanding of this common traffic phenomenon in a city-scale is still missing. An open question is how the traffic in the network collapses from a global efficient traffic to isolated local flows in small clusters, i.e. the question of traffic percolation. Here we study the traffic percolation properties on a lattice by simulation of an agent-based model for traffic. A critical traffic volume in this model distinguishes the free-state from congested state of traffic. Our results show that the threshold of traffic percolation decreases with increasing traffic volume and reaches a minimum value at the critical traffic volume. We show that this minimal threshold is the result of longest spatial correlation between traffic flows at the critical traffic volume. These findings may help to develop congestion mitigation strategies in a network view.
The urban transportation system, as one of the critical point for the phase transition from network connectivity to infrastructures, to a great extent, facilitates the performance in fragmentation. This threshold can be calculated theoretically many aspects of human activities. However, traffic transition, based on analytical considerations, or by simulations. from free flow to seriously congested state [1], occurs Recently, by constructing a traffic dynamical network, it is frequently in our daily life, deteriorates the system’s found that the organization of city traffic can be considered as efficiency, and even causes related problems such as pollution a percolation-like transition [11]. This result is based on and waste of time [2]. To date, various models based on analysis of city traffic real data and it was revealed that the different approaches have been proposed to simulate the city-scale global traffic is dynamically composed of different traffic flow and to achieve a comprehensive picture of the local traffic clusters, which evolve during the day. However, traffic dynamics including the features of traffic transition. the relation between traffic parameters (volume and routing These modelling methods [1, 3-8] mainly focus on two scales: choice) and the system percolation properties has not been macroscopic and microscopic. Macroscopic studies have been studied systematically. dedicated to reveal the traffic dynamics through the theory of In the present study, based on an agent-based model kinetic gas or fluid dynamic; on the other hand, the developed by Echenique et al. [12] and extensive simulations, microscopic study aims at capturing behaviors of the we study the relation between percolation organization and individual cars and simulating the basic interactions among traffic by varying two main traffic factors: the traffic volume them. Based on these methods, phenomena of jamming and its OD and the path-routing choice. It has been found [12] that relevance to systematic deterioration of traffic are extensively with the increase of traffic volume OD in the network, the studied. traffic will undergo a phase transition between free state and However, the mechanisms of how a network-scale global congested state at a critical value ODc. We introduce here a traffic disintegrates into local flows are still unclear. Indeed, percolation parameter p, which characterizes the efficiency of the global traffic system may be improperly improved due to the global traffic (for the detailed definition of p, see the the lack of knowledge on the interactions of local congestions section Traffic percolation below). We analyze here the and their influence on the macroscopic scale. We propose here critical threshold of traffic percolation pc as a function of OD to study the organization of global traffic flow on a lattice and find that pc has a sharp minimum value at the traffic model using percolation approach [9, 10]. Percolation theory transition threshold ODc, indicating the existence of the most is a useful tool for studying the network organization and efficient organization of global traffic flow. The value of pc is disintegration, with extensive applications as diverse as influenced by the routing choice. We reveal here that this diffusion, epidemics, and system robustness. In the behavior of pc results from the spatial correlation of traffic percolation theory, the threshold pc is defined as the critical load, which is maximal at traffic transition ODc. We also
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Percolation properties in a traffic model
analyzed the distribution of node betweenness in the giant k with minimal effective distance ekj will be selected as the component at percolation criticality. Finally, we discuss how packet’s next stop. In our simulation, if two or more paths our findings may provide insights to the practical management have identical effective distance, we randomly select one of of urban traffic. these paths. The routing scheme in Eq. (1) gives us a chance to capture the path-selection behavior of the drivers, which Traffic model. – Here, we focus on simulations of a traffic considers both the global knowledge of network’s topology dynamical model on a diluted periodical square lattice. In the and the current traffic situation ahead. Notice that a small h square lattice, links and nodes represent road sections and means that drivers care more about the local traffic situation their intersections respectively. To take into account the and are willing to circumvent the congested districts. disorder factors due to traffic control measures and diverse For each simulation of traffic flow in our study, we run road conditions, a certain fraction of randomly selected links the model for sufficient time steps to reach a stationary state, are initially removed to avoid a purely regular network and which is indicated by the system’s order parameter, ρ, defined weights are added to the remaining links. Specifically, in our as [12], study we randomly removed 10 percent of edges. Link LtLt() -( ) weights follow a Gaussian distribution G(μ, σ), where average lim NtN (2) t μ is 1 and standard deviation σ is 0.33 in the presented study. tOD Other values of σ can yield similar results. In our study, for simulating the adaptive traffic collective Where LN(t) is the total load in the network at time step t and behaviours, we use an agent-based model [12]. To simulate Δt is the observation time window. The infinite limit of t the traffic dynamics in the model, at each time step a given ensures that we reach a stationary state, where ρ becomes number (OD) of packets (cars) are created and delivered in the stable with time. As an indicator of traffic situation of the network following a given routing strategy. Both the origin network, ρ estimates the balance of the outflow and inflow. and destination of each packet are selected randomly from all The case ρ = 1 represents a completely congested traffic, since the nodes. The size of packet is chosen randomly following a no car can reach its destination, and ρ = 0 refers to a free-flow Gaussian distribution. For simplicity, the parameters of the state. Gaussian here are the same as the distribution of link weight. In the following, we focus on the features of traffic Each packet contributes to the load of its current node by the percolation as a function of traffic volume OD and the routing addition of its size. It is assumed that each node has an strategy h. identical capability to deliver only one packet at each time step, with the first-in-first-out basis. At each time step, the Traffic percolation. – To understand and reveal how the delivery process of packets on the network is carried out in a city-scale traffic flow breaks down into clusters of local flows, parallel update algorithm. After the delivery of all packets and we apply a percolation analysis on the model simulation the update of load configuration, we start a new iteration and results. At the stationary state of each traffic flow simulation, the time step t is increased by one. As the consideration of we record the load on every node. Then a threshold, m, which drivers’ capability to circumvent congested areas, a packet, varies from the minimum to the maximum of the node load, is setting off from its current site, selects its travel path defined to act as the only control parameter in our percolation dynamically according to a traffic-aware routing scheme analysis. The state si of the node with load li will be grouped discussed below and will be removed after reaching its into one of two classes: free-state for li ≤ m or congested for li destination. > m, i.e. The traffic-aware routing strategy [12] incorporates the 1, lmi local traffic awareness with global information in the delivery si (3) 0, lm algorithm, representing drivers’ behavior on path selection i according to the dynamical traffic situation. When a packet The fraction of free-state nodes (si = 1) in the network, with destination j staying on node i is choosing its next stop which can be considered as the occupation fraction p in among the i’s neighbors, an effective distance ekj between a percolation, increases as we raise the threshold m gradually. In neighbor k of i and j is introduced and defined as this process, we choose an appropriate interval (Δm = 0.002 in our study) so that the increase of m is small enough to keep at ekj hd kj(1- h ) l k (1) most one node’s state modified (one si changes from 0 to 1). Here, dkj is the optimal distance, defined here as the length of For nodes with equal load, we randomly select one of them path minimizing the sum of link weights between k and j, and update its state. As we increase the threshold m, clusters which is computed using Dijkstra algorithm [13, 14]. And lk is of free-state nodes (si = 1) will emerge and we can observe the the current load of node k, which is the total contribution occurrence of percolation process in the network: for small m, made by the size of all packets accumulated as a queue on the almost all nodes are considered as congested and only small node k. h is a tunable parameter between 0 and 1, that clusters will appear; for large m, small clusters will merge into accounts for the degree of traffic awareness incorporated in larger clusters, showing the organization process of local the delivery algorithm. Following this definition, the neighbor traffic flow. The emergence of the giant cluster indicates the
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Percolation properties in a traffic model occurrence of percolation transition and the formation of global flow can be efficiently maintained with minimal global traffic flow. At a certain fraction of free-state nodes pc number of local flows at criticality. We demonstrate that the (determined by mc), the second-largest cluster reaches its results are similar under different path-selecting behaviors maximum, which signifies the formation of global traffic flow represented by h (fig. 2b), with different value of minimal pc. [9] (fig. 1a and 1c). According to percolation theory, this value Note also, that only near ODc the deviation from random pc refers to the critical threshold of traffic percolation [9]. This percolation is significant and reaches the maximal value at clustering process is illustrated in fig. 1b and 1d. ODc. When we recover the classical random percolation by shuffling the node loads, the interesting non-monotonic
1.0 behavior and the sharp minimum of pc disappear. We next a study the origin of this striking minimum of pc in the traffic 0.8 G organization. 0.6 SG
Size h = 0.7 0.4 1.0 1.0 a 0.65 b 0.2 0.6
0.60
c
ρ ρ
0.0 c
p
0.5 p 0.0 0.2 0.4 0.6 0.8 1.0 0.5 p p pc 0.5 c 0.55 1.0 p (Random) p (Random) c c c ρ h = 0.7 0.50 ρ h = 0.3 0.8 0.4 0.0 0.0 G 0 50 100 150 200 250 300 10 20 30 40 50 60 70 80 90 100 0.6 SG OD OD
Size h = 0.3 0.4 Fig. 2: The critical threshold pc and order parameter ρ as a 0.2 function of OD for different h (0.7 and 0.3 in (a) and (b) respectively). The critical OD, ODc, are near 120 and 55 for h 0.0 0.0 0.2 0.4 p 0.6 0.8 1.0 = 0.7 and 0.3 respectively. For comparison, we show the classical random percolation (in dash line) by shuffling the Fig. 1: Percolation of traffic flow on the lattice. In (a) and (c), node loads. Simulations have been carried out on a lattice of we show the size of both the largest functional cluster G and size 100×100 with averages over 200 realizations. We choose the second-largest cluster SG changes with p for different h: our lattice with linear size L = 100, because bigger system (a) h = 0.7 and (c) h = 0.3. Note that we define here G as the sizes are very time consuming. size of largest cluster among all clusters in the diluted lattice obtained for a given m. In (b) and (d), we show the Correlation length in traffic. – Correlations have been percolation clusters of the network near pc for different h: (b) h = 0.7 and (d) h = 0.3. The largest cluster is marked in olive found between traffic flow in highways [1]. To understand green and the second largest one is marked in orange. All the how the traffic volume influences pc, in particular the origin of grey nodes are those congested and removed from the strong deviation from random percolation near ODc, we network. Other colors refer to other small functional clusters. investigate the spatial correlation length [18-21] of the traffic The results here are obtained on a lattice (100×100) at OD = load. Different from the random percolation, the local flows 150. interact with correlations. It is interesting to study how the correlation of traffic flows influences the threshold pc of
traffic percolation. We calculate the spatial correlation of In a real urban transportation system, the dynamical loads with different OD in the network. The correlation traffic volume is a parameter of great importance, because of function, C(r), measures the correlation between traffic loads its direct effect on the emergence of congestions [1, 15-17]. at distance r [19-21]: To observe the effect of traffic volume on traffic percolation, we first analyze the order parameter ρ, Eq. (2), of the system ()()xxxxrr () 1 ij， i S ijij at the stationary state as a function of traffic volume, OD. We Cr( ) (4) 2 ()rr can observe a transition at ODc from free traffic to congestion ij， i S ij as OD increases (fig. 2a and 2b) , similar to that found in [12]. A critical traffic volume ODc can be found, above which Here, xi is the load on node i. x is the average load over all systematic congestions occur. We next study how these traffic nodes and 22()xxN is the variance. Here, i， i S iS parameters affect the traffic percolation properties. S is the set of all nodes in the network of size NS. rij is the In the analysis of the traffic percolation properties, pc Euclidean distance between nodes i and j. The delta function can be regarded as a quantitative measure for the efficiency of selects nodes at distance r. Following this definition, positive the global traffic flow [11]. In the present study, we explore values of C(r) mean that traffic situations at distance r are the influence of traffic volume on pc for different OD (fig. 2a positive correlated, and vice versa. and 2b). Strikingly, our findings suggest that pc initially We show in fig. 3a and 3b that the correlation decays in decreases non-linearly with increasing OD. A sharp minimum different way for different traffic volume. The correlation pc is clearly observed at the critical OD, indicating that the length ξl, is defined as the distance, where C(r) becomes zero
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Percolation properties in a traffic model for the first time [19], representing the scale of the correlated a -1 h = 0.7 -1 b h = 0.3 domains (fig. 3a and 3b). We show the correlation length as a 10 10 -2
function of OD values in fig. 3c and 3d, and find that the 10-2 10 )
) -3 C -3 C
10 ( 10 (
correlation length increases when critical traffic volume, ODc, OD = 30 IIC IIC
OD = 90 -4 -4 10 10 OD = 55 is approached. The correlation length of traffic load reaches its OD = 120 -5 -5 10 OD = 150 10 OD = 80 maximum at ODc for different h, suggesting the traffic flow is -6 -6 10 -3 -2 -1 0 10 -3 -2 -1 correlated with the longest range at the critical point. 10 10 10 10 10 10 10 C C
0.8 c h = 0.7 0.018 d h = 0.3 a OD = 90 0.3 b OD = 30 0.03
0.6 OD = 120 OD = 55 IIC
IIC
> > OD = 150 0.2 OD = 80 C
) 0.4
C
)
<
r r
(
( <
0.015
C h = 0.7 C h = 0.3 0.2 0.1 0.02 0.0 0.0 -0.2 0.012 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 60 80 100 120 140 160 30 40 50 60 70 80 r r OD OD 20 c h = 0.7 d h = 0.3 Fig. 4: The distribution of the betweenness centrality in the 15 Random 10 Random giant component. (a) and (b) demonstrate the distribution of
l
l betweenness, PIIC(C), for different OD. In (c) and (d), we
10 5 show the averaged betweenness of nodes in the giant 5 component
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Percolation properties in a traffic model
[10] BUNDE A. and HAVLIN S., Fractals and In conclusion, by capturing some realistic behaviors of disordered systems (Springer Science & Business drivers in the daily traffic using an agent-based model Media, Germany) 2012. featured with a traffic-aware routing strategy, we investigate [11] LI D., FU B., WANG Y., LU G., BEREZIN Y., the properties of traffic percolation under the influence of two STANLEY H.E. and HAVLIN S., Proc. Natl. traffic factors: the traffic volume and the drivers’ path-routing Acad. Sci. U.S.A., 112 (2015) 669. choice. Interestingly, we find that the maximal efficiency of [12] ECHENIQUE P., GOMEZ-GARDENES J. and traffic occurs at ODc, where traffic percolation has a minimal MORENO Y., Europhys. Lett., 71 (2005) 325. threshold pc. We show that this results from the enhanced [13] BRAUNSTEIN L.A., BULDYREV S.V., strength of spatial correlation of loads at criticality. At the COHEN R., HAVLIN S. and STANLEY H.E., presence of the relationship between percolation and traffic, Phys. Rev. Lett., 91 (2003) 168701. we reveal that the high accumulation of optimal paths in a [14] DIJKSTRA E.W., Numer. Math., 1 (1959) 269. giant percolation component provides a possible routing [15] KERNER B.S. and REHBORN H., Phys. Rev. choice to avoid congestions, which may suggest ways of Lett., 79 (1997) 4030. mitigating the traffic congestions. The particular definition of [16] KERNER B.S., The physics of traffic: empirical effective distances [12] is essential to capture the realistic freeway pattern features, engineering applications, behaviors of drivers in the daily traffic. Given that the linear and theory (Springer, Germany) 2012. dependence of routing strategy in Eq. (1) is assumed, a study [17] ZHAO L., LAI Y.-C., PARK K. and YE N., Phys. is needed in the future to find out the dependence function and Rev. E, 71 (2005) 026125. its effect on traffic percolation from real traffic data. [18] MAKSE H.A., HAVLIN S., and STANLEY H.E., Furthermore, inspired by the successful application of Nature, 377 (1995) 608. network-view approaches to the study of realistic complex systems [28-30], we expect the future exploration on the [19] CAVAGNA A., CIMARELLI A., GIARDINA I., comparison of our findings with the analysis from real data. PARISI G., SANTAGATI R., STEFANINI F. and VIALE M., Proc. Natl. Acad. Sci. U.S.A., 107 (2010) 11865. *** [20] GALLOS L.K., BARTTFELD P., HAVLIN S., This work is supported by Collaborative Innovation SIGMAN M. and MAKSE H.A., Sci. Rep., 2 Center for industrial Cyber-Physical System and the National (2012) 454. Basic Research Program of China (2012CB725404). D.L. [21] DAQING L., YINAN J., RUI K. and HAVLIN S., acknowledges support from the National Natural Science Sci. Rep., 4 (2014) 5381. Foundation of China (Grant 61104144) and the Fundamental [22] PRAKASH S., HAVLIN S., SCHWARTZ M. Research Funds for the Central Universities. S.H. thanks and STANLEY H.E., Phys. Rev. A, 46 (1992) Defense Threat Reduction Agency (DTRA), Office of Naval R1724. [23] MAKSE H.A., HAVLIN S., SCHWARTZ M. Research (ONR) (N62909-14-1-N019), United States-Israel and STANLEY H.E., Phys. Rev. E, 53 (1996) 5445. Binational Science Foundation, the LINC (Grant 289447) and [24] NEWMAN M.E.J., Phys. Rev. E, 64 (2001) the Multiplex (Grant 317532) European projects and the Israel 016132. Science Foundation for support. [25] WU Z., BRAUNSTEIN L.A., HAVLIN S. and STANLEY H.E., Phys. Rev. Lett., 96 (2006) REFERENCES 148702. [1] HELBING D., Rev. Mod. Phys., 73 (2001) 1067. [26] SREENIVASAN S., KALISKY T., [2] SWUTC., 2012 Annual Urban Mobility Report. BRAUNSTEIN L.A., BULDYREV S.V., https://mobility.tamu.edu/ums/, 2012. HAVLIN S., and STANLEY H.E., Phys. Rev. E, [3] LIGHTHILL M.J. and WHITHAM G.B., Proc. R. 70 (2004) 046133. Soc. London, Ser. A, 229 (1955) 281. [27] LI G., BRAUNSTEIN L.A., BULDYREV S.V., [4] PRIGOGINE I. and HERMAN R., Kinetic Theory HAVLIN S., and STANLEY H.E., Phys. Rev. E, of Vehicular Traffic (Elsevier, New York) 1971. 75 (2007) 045103. [5] BANDO M., HASEBE K., NAKAYAMA A., [28]BOERS N., BOOKHAGEN B., BARBOSA H., SHIBATA A. and SUGIYAMA Y., Phys. Rev. E, MARWAN N., KURTHS J., and MARENGO J., 51 (1995) 1035. Nat. Commun., 5 (2014) 5199. [6] TREIBER M., HENNECKE A. and HELBING D., [29] MENCK P.J., HEITZIG J., KURTHS J., and Phys. Rev. E, 62 (2000) 1805. JOACHIM SCHELLNHUBER H., Nat. [7] NAGEL K. and SCHRECKENBERG M., J. Phys. Commun., 5 (2014) 3969. I, 2 (1992) 2221. [30] ZOU W., SENTHILKUMAR D., NAGAO R., [8] HELBING D. and HUBERMAN B.A., Nature, KISS I.Z., TANG Y., KOSESKA A., DUAN J., 396 (1998) 738. and KURTHS J., Nat. Commun., 6 (2015) 8709. [9] COHEN R. and HAVLIN S., Complex networks: structure, robustness and function (Cambridge
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