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# Macroscopic dynamics and the collapse of urban traffic

Edited by Paul Trunfio, Boston University, Boston, MA, and accepted by Editorial Board Member Pablo G. Debenedetti June 14, 2018 (received for review February 15, 2018)

## Abstract

Stories of mega-jams that last tens of hours or even days appear not only in fiction but also in reality. In this context, it is important to characterize the collapse of the network, defined as the transition from a characteristic travel time to orders of magnitude longer for the same distance traveled. In this multicity study, we unravel this complex phenomenon under various conditions of demand and translate it to the travel time of the individual drivers. First, we start with the current conditions, showing that there is a characteristic time τ that takes a representative group of commuters to arrive at their destinations once their maximum density has been reached. While this time differs from city to city, it can be explained by Γ, defined as the ratio of the vehicle miles traveled to the total vehicle distance the road network can support per hour. Modifying Γ can improve τ and directly inform planning and infrastructure interventions. In this study we focus on measuring the vulnerability of the system by increasing the volume of cars in the network, keeping the road capacity and the empirical spatial dynamics from origins to destinations unchanged. We identify three states of urban traffic, separated by two distinctive transitions. The first one describes the appearance of the first bottlenecks and the second one the collapse of the system. This collapse is marked by a given number of commuters in each city and it is formally characterized by a nonequilibrium phase transition.

The steady increase of traffic congestion not only translates into overpowering travel times (1⇓–3), but also erodes economic growth (4) and has negative environmental impacts (5⇓–7). Urban traffic has extensively been studied through computer simulations with particular interest in the characterization of the transition from free flow to congestion (8⇓–10). Traditional approaches in traffic engineering have described the phenomenon within the framework of a well-defined relationship between network-wide average flow and the density of cars (11⇓⇓⇓⇓⇓–17). Considering the network density as the control parameter, a congested state at the network level emerges when car outflow starts decreasing when the vehicular demand exceeds a certain value. In parallel, critical loads

The recent availability of data on personal tracking devices has enriched the study of traffic models. Origin–destination (OD) tables can be extracted from call-detailed records (CDRs) of mobile phones (21, 22) and GPS-equipped vehicles can act as sensors of traffic conditions (15, 23). Patterns of individual mobility have been uncovered 24, 25) and allow us to model individual daily mobility from passively collected sources (26). Comparing various cities, scaling of urban indicators emerges (27, 28). Examples are travel times and road network characteristics as a function of population and socioeconomic characteristics (27, 29, 30). For operational and planning purposes, a macroscopic description of the urban traffic dynamics and their vulnerability to collapse, measured in terms of car volumes, road network supply, and individual travels, is essential, yet still missing. In other words, In what way does the information contained in the ODs determine the travel time of target individuals and how can these dynamics be understood in terms of actionable quantities to explain when the system will collapse?

As a first step in that direction, Çolak et al. (22) used a framework of static equilibrium to compare the morning conditions of congested travel times (

First we directly measure the reported times of individual travel diaries in Boston, San Francisco, and Bogotá, and we build their reported temporal profile of car trips vs. time in the morning peak period. We focus on the target group of vehicles that enter the network within the peak hour (7:00–8:00 AM) and find that they have an exponential unloading period. This empirical observation is corroborated by a calibrated and validated simulation-based dynamic traffic assignment model of the entire city of Melbourne. We argue that this characteristic unloading time τ faced by commuters can be related by macroscopic characteristics that contain the overall road capacity and travel demand. To uncover the macroscopic dynamic that explains τ, we implemented a cellular automata (CA) model on the road networks of five cities around the world, namely Boston, Porto, Lisbon, Rio de Janeiro, and the San Francisco Bay area. Informed by the empirical trip demand derived from CDRs from each city, we analyze the morning peak hour by loading constantly during 1 h and letting the drivers arrive at their destination. We show that the exponential form of the unloading time is a consequence of the log-normal distribution exhibited by the commuting distances. As expected, *A* for Boston. We show that the dynamical response is independent of the level of detail of the traffic model and the city under consideration.

## The Unloading Time of Urban Road Networks

Peak-hour traffic congestion is inherent to the circadian rhythm of modern city life. After an intense and heterogeneous loading period, there is a recovery period where traffic jams on major arterial roads dissipate, thus relieving the city traffic. We focus, in this work, on the recovery period after the morning peak hours under current traffic conditions and further see how it changes as a function of the number of cars in diverse cities, paying particular attention to the collapse of the system.

We start by gathering empirical observations on urban congestion. We analyze individual travel diaries in Boston, San Francisco, and Bogotá. This allows us to build an approximate temporal profile of car trips in the network. To have a controlled sample, we focus on the target group of vehicles that enter the network within the peak hour, defined as the hour immediately before the peak, here from 7:00 AM to 8:00 AM. Remarkably, we observe a generic behavior: After a loading period, the number of target group vehicles still in the network, *B*. It is important to note that this observation is in contrast with that of all of the vehicles of commuters also shown in Fig. 1*B*. This sample has a more heterogeneous mix of times to unload, not properly characterized by a single relaxation time.

We further measure the loading and unloading with the simulation-based dynamic traffic assignment (DTA) model of the city of Melbourne (32). This model has been carefully calibrated for the entire city and simulates almost 2.1 million commuters in a 4-h morning peak period (*Materials and Methods*). Similar to the empirical results reported in surveys, we observe a recovery period that follows an exponential decay that can be written as*C*). This implies a proportionality between the exit function *SI Appendix*, Fig. S2). Thus, τ should indicate the network response to the congestion, including also vehicles not belonging to the target group. We further study how τ depends on network characteristics and different travel demands over diverse cities.

## CA Model

Due to the complexities of large-scale traffic simulations, to compare cities we implement a CA model (33). As an input, we use validated travel demand models obtained by Çolak et al. (22). We focus on traffic demand from 7:30 AM to 8:30 AM for the subject cities (22) (*Materials and Methods*). To model the observed recovery, we load the road network during 1 h; i.e., every time step

The initial route in the road networks is precalculated with the congested traveled time *A*).

We implement the deterministic Nagel–Schreckenberg CA model (33), where for simplicity each edge only has one lane. When a car is traveling in free flow and approaches an intersection, it decreases its speed to

In each time step in a random sequence, the intersections transmit the first vehicles of the originating streets. The vehicles pass through a street segment (*i*) if the first cell in the next desired street is empty and (*ii*) if the road capacity of the originating street allows it. In the latter case, the vehicle is delivered with a probability p proportional to the road capacity of the originating street, *SI Appendix*, Table S1). In the case of long waiting times, we introduce a basic dynamic routing strategy, commonly known as adaptive driving: A vehicle that has been stopped at an intersection during more than

In *SI Appendix*, Fig. S3, we show how this simple model offers a reasonable description of important empirical features of urban traffic reported in previous works (12, 15, 17).

## Comparing Morning Traffic

Due to the one-lane representation of the streets and the simplicity of the vehicle dynamics, the networks studied via the CA model are more susceptible to congestion if using the empirical volume demand (V). We thus rescaled V by the ratio between the space demands in simulated and real networks, given by*SI Appendix*, Fig. S4). Despite the differences in road infrastructure, there are small differences between these ratios for the considered cities, as shown in Table 1.

Next, we distribute the calibrated volume demand

The observed patterns here are independent of the simulation modeling methodology. Both the simplified simulation and the more complex DTA model reveal similar outcomes. Fig. 2*B* shows the loading and unloading in terms of the fraction of vehicles in the network, *B*, *Inset*). As expected, we can see the network responds differently from city to city. Note that the values of

The exponential recovery can be explained by commuting trip distances, d, which can be approximated by a log-normal distribution (*SI Appendix*, Fig. S5), *C*, *Left*). In a noncollapsed state, the remaining travel times, *C*, *Right*. Thus, in the recovery period after the first hour, the number of vehicles as a function of time, *SI Appendix*, Fig. S6).

We can further explain *D* there is an increasing relation between *SI Appendix*, Fig. S9 shows a sensitivity analysis of

## Dynamics of Urban Traffic

To have a complete understanding of τ and the macroscopic dynamics of urban traffic, we analyze different demand levels, keeping the spatial distribution of trips and the loading in the morning peak hour. In doing so, we uncover three different states of urban traffic (Fig. 3). We call them free flow, traffic jam, and network collapse, also known as gridlock. We further use the superscript i to emphasize that the values are calculated for each city i. First, for very low *C*, indicating that after their particular threshold *SI Appendix*, Fig. S10). Now, if the demand keeps growing, long-lasting traffic jams emerge, deforming the initially exponential unloading, and thus τ increase dramatically as shown in Fig. 3*C*. This is the onset of the transition to the network collapse, because for a large number of cars we do not observe an exponential unloading anymore. From this state, τ is measured as how long it takes to unload the network to reach the value of

Γ is a state variable that allows us to compare the congestion level for different cities. Interestingly, according to its definition (Eq. **5**), Γ is an extensive variable, and thus we can expect a dependency on the city size. *SI Appendix*, Fig. S11 suggests that *P*) and spatial extent of the urban area (*A*). We find

## Urban Vulnerability and Transition to a Collapsed State

While the considered cities already face high traffic demand, the studied exponential decay indicates a characteristic time in the recovery without the occurrence of long-lasting traffic jams that expand to the majority of the network. We further compare the emergence of collapse induced by the number of cars.

Fig. 4*A* shows the recovery period for all five cities under high loading rates. At a certain critical value *SI Appendix*, Fig. S12. In this transient collapsed state, most of the vehicles remain trapped in long-lasting gridlocks. Eventually, due to the rerouting possibilities, the system recovers and unloads completely.

The dynamics of *B* shows, we find that curves collapse when

We further study the transition in the supercritical region, using the loading rate as a control parameter. The natural order parameter is the remaining percentage of vehicles in the network at very long times, *A* (also *SI Appendix*, Fig. S13).

Fig. 5*A* shows *A*, *Inset*. Fig. 5*B* shows how β increases with increasing α. For comparison, the universal exponents of (1 + 1) and (2 + 1) dimensions in DP are also sketched.

To compare the response of various cities, we define *C* shows that this quantity diverges at the vicinity of

The dynamics of the presented results do not depend on the rules of the CA. A sensitivity analysis (*SI Appendix*, Fig. S14) indicates that, as expected, *SI Appendix*, Figs. S15 and S16, we study the percolation transition with various demand volumes. From the perspective of the nonequilibrium phase transitions, future work should focus on studying the spatial and temporal correlation lengths around the criticality (*SI Appendix*, section A.

## Discussion

We have uncovered an exponential recovery in urban traffic, given by the unloading time of a target group in the peak hour. This value relates individual mobility with traffic congestion levels and depends on the road infrastructure and travel demand. Within this framework, we measure the vulnerability of urban networks to loading vehicles, while keeping the trip distributions and street capacities unchanged. We show that the transition to urban gridlock resembles the DP universality class, and it is studied as a nonequilibrium phase transition. In a data-rich reality, the aim of this work is to open avenues for direct measurements and description of urban traffic with methods of statistical physics.

## Materials and Methods

### Surveys Datasets.

Data were derived from the 2010/2011 Massachusetts Travel Survey (MTS) and the 2000 Bay Area Transportation Survey (BATS).

### Extraction of Validated Travel Demand Information.

Travel information can be extracted from the analysis of CDRs from mobile phones. Çolak et al. (22) estimated and validated the OD tables during the morning peak hour for the same five cities discussed here. Using these OD tables, along with road networks publicly available on OpenStreetMaps (OSM), the congested travel time (

### Simulations of Melbourne.

DynaMel is a large-scale simulation-based DTA model of Melbourne, Australia. The model has been calibrated and validated to simulate the 6:00 AM to 10:00 AM morning peak period. The model consists of 55, 719 links and 24, 502 nodes and simulates almost 2.1 million commuters in the 6:00 AM to 10:00 AM morning peak period. As an input of the DTA model, DynaMel applies a machine-learning–based technique to calibrate the traffic-flow fundamental diagram using the observed traffic data from hundreds of freeway loop detectors across the entire network.

### Details of the CA Model.

Vehicle dynamics along road segments are modeled with the deterministic Nagel–Schreckenberg CA model (33). For simplicity, each edge has only one lane. Therefore, every road segment is discretized in cells of equal length, *A*). When a car is traveling in free flow and approaches an intersection, it decreases its speed to

For contractual and privacy reasons, we cannot make the raw data available. We are pleased to make available the data of the OD matrices, software to replicate the method, and the appropriate documentation. This information may be accessed at the GitHub repository https://github.com/leolmoss/CollapseUrbanTraffic. This repository is sufficient to reproduce the results of this paper.

## Acknowledgments

This work was supported by the Massachusetts Institute of Technology Energy Initiative (MITEI), and the MIT Environmental Solutions Initiative as well as by Lawrence Berkeley National Laboratory’s Laboratory Directed Research and Development (LDRD) funds. Views and conclusions in this document are those of the authors and should not be interpreted as representing the policies, either expressed or implied, of the sponsors.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: martag{at}berkeley.edu.

Author contributions: L.E.O. and M.C.G. designed research; L.E.O. and M.C.G. performed research; L.E.O., S.Ç., S.S., M.S., and M.C.G. analyzed data; and L.E.O. and M.C.G. wrote the paper.

The authors declare no conflict of interest.

This paper results from the Arthur M. Sackler Colloquium of the National Academy of Sciences, “Modeling and Visualizing Science and Technology Developments,” held December 4–5, 2017, at the Arnold and Mabel Beckman Center of the National Academies of Sciences and Engineering in Irvine, CA. The complete program and video recordings of most presentations are available on the NAS website at www.nasonline.org/modeling_and_visualizing.

This article is a PNAS Direct Submission. P.T. is a guest editor invited by the Editorial Board.

Data deposition: The data of the OD matrices, software to replicate the method, and the appropriate documentation can be accessed on GitHub at https://github.com/leolmoss/CollapseUrbanTraffic.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1800474115/-/DCSupplemental.

Published under the PNAS license.

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