A critical phenomenon is an intrinsic feature of traffic dynamics, during which transition between isolated local flows and global flows occurs. However, very little attention has been given to the question of how the local flows in the roads are organized collectively into a global city flow. Here we characterize this organization process of traffic as "traffic percolation," where the giant cluster of local flows disintegrates when the second largest cluster reaches its maximum. We find in real-time data of city road traffic that global traffic is dynamically composed of clusters of local flows, which are connected by bottleneck links. This organization evolves during a day with different bottleneck links appearing in different hours, but similar in the same hours in different days. A small improvement of critical bottleneck roads is found to benefit significantly the global traffic, providing a method to improve city traffic with low cost. Our results may provide insights on the relation between traffic dynamics and percolation, which can be useful for efficient transportation, epidemic control, and emergency evacuation. emergence | percolation | traffic T raffic, as a large-scale and complex dynamical system, has attracted much attention, especially on its dynamical transition between free flow and congestion (1-3). The dynamics of traffic have been studied using many types of models (4-11), ranging from models in macroscopic scales based on the kinetic gas theory or fluid dynamics to approaches in microscopic scales with equations for each car in the network. However, there is still a gap between the microscopic behavior of individual vehicles and the emergence of macroscopic city traffic. Indeed, a fundamental question has rarely been addressed: how the local flows in roads interact and organize collectively into global flow throughout the city network. This knowledge is not only necessary to bridge the gap between local traffic and global traffic, but also essential for developing efficient traffic control strategies. There are mainly two obstacles in studying how the collective network dynamics of real traffic emerge from local flows. The first obstacle is the lack of valid methods to quantify the dynamical organization of traffic in the road network. The second is the lack of data on traffic dynamics in a network scale. To overcome the first obstacle, we develop here a quantitative framework based on percolation theory, which combines evolving traffic dynamics with network structure. In this framework, instead of the commonly used structural topology, only roads in the network with speed larger than a variable threshold are considered functionally connected. In this way, we can characterize and understand the formation process of traffic dynamics. To overcome the second obstacle of missing data on a network scale and understand the organization processes of real traffic in a network, we collected and analyzed velocities of more than 1,000 roads with 5-min segments records measured in a road network in a central area of Beijing (Fig. 1A) . This area of more than 22 km 2 contains the largest train station in Beijing and is considered a typical region showing transition between free flow and congestions. The data cover a time span of 2 wk in 2013. For the road network, nodes represent the intersections and edges represent the road segments between two intersections. For each road, the velocity v ij (t) varies during a day according to real-time traffic. For each road e ij , we set the 95th percentile of its velocity in each day as its limited maximal velocity and define r ij (t) as the ratio between its current velocity and its limited maximal velocity measured for that day (Fig. 1B and SI Appendix, Fig. S1 ). For a given threshold q, the road e ij can be classified into two categories: functional when r ij > q and dysfunctional for r ij < q, e ij = 1; r ij ≥ q 0; r ij < q: In this way, a functional traffic network can be constructed for a given q value from the traffic dynamics of the original road network, which becomes more diluted as the value of q increases. To observe the emergence of global city traffic in the network scale at a given time, we can vary the value of q and study the formation process of the dynamical traffic network. For q = 0, the traffic network is the same as the original road network and for q = 1 it becomes completely fragmented. For a certain value of q, the hierarchical organization of traffic in different scales emerges, where only clusters of roads with r ij higher than q appear (clusters in Fig. 2 A-C). These clusters represent functional modules composed of connected roads with speed higher than q. For example, during a typical lunchtime instant, for q = 0.69, as The transition between free flow and congestions in traffic can be observed in our daily life. Although this traffic phenomenon is well studied in highways, traffic in a network scale (representing a city) is far from being understood. A fundamental unsolved question is how the global flow in a city is being integrated from local flows. Here, we identify a fundamental mechanism of traffic organization in a network scale as a percolation process, and we show how global traffic breaks down when identified bottlenecks are congested. These bottlenecks evolve with time according to traffic dynamics and are different from structural bottleneck links found by traditional network analysis. Improvement of traffic on these bottlenecks can significantly improve the global traffic.
According to the research in REF , the speed ratio can describe the congestion situation for a road.
Percolation transition in dynamical traffic network with evolving critical bottlenecks.
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America
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