Abstract In human crowds as well as in many animal societies, local interactions among individuals often give rise to self-organized collective organizations that offer functional benefits to the group. For instance, flows of pedestrians moving in opposite directions spontaneously segregate into lanes of uniform walking directions. This phenomenon is often referred to as a smart collective pattern, as it increases the traffic efficiency with no need of external control. However, the functional benefits of this emergent organization have never been experimentally measured, and the underlying behavioral mechanisms are poorly understood. In this work, we have studied this phenomenon under controlled laboratory conditions. We found that the traffic segregation exhibits structural instabilities characterized by the alternation of organized and disorganized states, where the lifetime of well-organized clusters of pedestrians follow a stretched exponential relaxation process. Further analysis show that the inter-pedestrian variability of comfortable walking speeds is a key variable at the origin of the observed traffic perturbations. We show that the collective benefit of the emerging pattern is maximized when all pedestrians walk at the average speed of the group. In practice, however, local interactions between slow- and fast-walking pedestrians trigger global breakdowns of organization, which reduce the collective and the individual payoff provided by the traffic segregation. This work is a step ahead toward the understanding of traffic self-organization in crowds, which turns out to be modulated by complex behavioral mechanisms that do not always maximize the group's benefits. The quantitative understanding of crowd behaviors opens the way for designing bottom-up management strategies bound to promote the emergence of efficient collective behaviors in crowds.

Author Summary A crowd of pedestrians is a complex system that exhibits a rich variety of self-organized collective behaviours. For instance, when two flows of people are walking in opposite directions in a crowded street, pedestrians spontaneously share the available space by forming lanes of uniform walking directions. This “pedestrian highway” is a typical example of self-organized functional pattern, as it increases the traffic efficiency with no need of external control. In this work, we have conducted a series of laboratory experiments to determine the behavioral mechanisms underlying this pattern. In contrast to previous theoretical predictions, we found that the traffic organization actually alternates in time between well-organized and disorganized states. Our results demonstrate that this unstable dynamics is due to interactions between people walking faster and slower than the average speed of the crowd. While the traffic efficiency is maximized when everybody walks at the same speed, crowd heterogeneity reduces the collective benefits provided by the traffic segregation. This work is a step ahead in understanding the mechanisms of crowd self-organization, and opens the way for the elaboration of management strategies bound to promote smart collective behaviors.

Citation: Moussaïd M, Guillot EG, Moreau M, Fehrenbach J, Chabiron O, Lemercier S, et al. (2012) Traffic Instabilities in Self-Organized Pedestrian Crowds. PLoS Comput Biol 8(3): e1002442. https://doi.org/10.1371/journal.pcbi.1002442 Editor: Eshel Ben-Jacob, Tel Aviv University, Israel Received: October 3, 2011; Accepted: February 8, 2012; Published: March 22, 2012 Copyright: © 2012 Moussaïd et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This study was supported by a research grant from the PEDIGREE project funded by the French National Research Agency (Grant No ANR-08-SYSC-015). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction In many biological and social systems, such as fish schools, ant colonies, or human crowds, repeated local interactions among individuals support the emergence of a variety of collective patterns of motion [1]–[4]. Under certain conditions, the emerging organization allows the group to solve efficiently coordination problems without centralized planning or external control. In human crowds, such functional patterns of motion have been identified many times in the past, such as the alternating flows at a bottleneck [5], the formation of trails [6], or the walking configuration of social groups [7]. Remarkably, nobody orchestrates these phenomena and pedestrians do not actively seek these emerging collective organizations. Instead, individuals behave according to their own motivations, but local interactions generate functional organizations at the scale of the crowd. Therefore, these phenomena are often considered as prime examples of collective intelligence, sometimes called “the wisdom of crowds” [8]–[10]. One of the well-known example of such functional self-organization in crowds is the formation of lanes in bidirectional flows [11]–[13]: When two flows of pedestrians are moving in opposite directions in a crowded street, people spontaneously share the available space by forming a “pedestrian highway”, where individuals walking in opposite direction segregate into lanes. This self-organized pattern of motion enhances the traffic flow by reducing frictional effects, local accelerations, energy consumptions and walking delays [14]. According to previous modeling works, the formation of lanes goes along with a sudden transition from disorder (where individuals are randomly distributed) to order (where opposite flows are segregated) [3]. Similar transitions from disorder to order have been observed in a wide variety of complex systems composed of locally interacting agents, in physical [15], [16], biological [2], [17]–[19] and social systems [10], [20], [21]. In human crowds, however, little is known about this phenomenon. From an empirical and quantitative point of view, the features of the spontaneous traffic organization remain scarcely documented, and the behavioral mechanisms underlying this phenomenon are hardly understood. In fact, it is poorly known how the transition operates, how the traffic organization evolves in time, and how much this collective organization benefits to the group. In this work, we have investigated the dynamics of lane formation under laboratory conditions, and studied the benefits provided by this traffic organization at the individual and crowd levels. To study the formation of lanes under experimental conditions, one major issue arising from past works is to handle the participants' inflow without interfering with the phenomenon [13], [22]. In fact, in a straight corridor, the starting positions of pedestrians regularly introduced from both ends strongly influence the resulting traffic organization, which is detrimental to the relevance of the measurements. To avoid this drawback, we have used a ring-shaped corridor that provides periodic boundary conditions [18], [23]. In this way, observing the phenomenon without perturbations induced by the experimental procedure becomes possible. To observe the emergence and the temporal dynamics of lane formation, N participants were randomly distributed in the ring-shaped corridor. A walking direction was randomly attributed to each of them, in such a way that N/2 participants walked clockwise and N/2 anti-clockwise. At the starting signal, participants started to walk in their attributed direction, allowing us to observe and characterize the emergence of traffic organization. A total of 11 replications were analyzed, with N = 60, 50 and 30 participants (3, 2 and 6 replications, respectively). In the following, we present our experimental results and show that the complex dynamics of traffic self-organization is based on simple behavioral mechanisms, where interactions between pedestrians walking faster and slower than the average trigger local perturbations that rapidly change into global traffic instabilities. While lane formation can be theoretically very efficient and functional, we show that, in practice, inter-individual variability undermines the overall benefits of the collective organization.

Discussion The spontaneous traffic organization of pedestrian flows is a functional self-organized collective pattern in human crowds, where people spontaneously share the available space by forming lanes of uniform walking direction. Based on experimental measurements, we found that this phenomenon exhibits structural instabilities, where mixed and well-segregated phases alternate in time. Our study demonstrates that speed variability among individuals is a key element underlying the observed traffic perturbations. Previous modeling work have suggested a similar relation between traffic stability and the fluctuations or the heterogeneity of the system, but these results were based on numerical simulations only [27], [28]. In particular, our data allowed us to unravel the precise mechanisms underlying the emergence of traffic perturbations: people walking slower create density gaps, while those walking faster make use of these gaps to overtake other pedestrians in front of them. These specific local interactions finally result in large-scale traffic breakdowns, and the spontaneous self-organization ends up in a sub-optimal state. Therefore, the collective payoff of the group is undermined because pedestrians try to increase their individual level of satisfaction. Indeed, it is known that walking at the comfortable walking speed provides individual metabolic-related benefits [29]. But even pedestrians who cooperate by walking at the average group speed are increasingly less satisfied as other individuals deviating from the average speed are numerous. This incompatibility between individuals' satisfaction and crowd payoff is typical of many social dilemmas where self-interest conflicts with group interest [30], [31]. Nevertheless, the functional benefit of traffic segregation is maximized in homogeneous crowds. Only diversity reduces the efficiency of the spatial self-organization. Many other decentralized systems facing coordination problems display the same trend. In car traffic, the variability among drivers' behaviors also lead to disturbing collective patterns, such as stop-and-go waves and large-scale traffic jams [9], [32], [33]. In other biological systems such as animal swarms, goal oriented collective motion is also disturbed by the presence of inter-individual variability [18], [34]. Remarkably, when facing other kinds of tasks, inter-individual variability may have the opposite effect and promote the emergence of efficient behaviors [35]–[37]. In collective decision-making problems, heterogeneity favors the discovery of new solutions and prevents the group from staying stuck in suboptimal behaviors [38], [39]. Therefore, it seems that group diversity can either promote or disturb collective intelligence depending on the nature of the task. Among the rich variety of self-organized collective behaviors observed in human crowds, not all of them offer functional benefits to the group. While some phenomena like traffic segregation, or alternating flows at bottlenecks provide decentralized solutions to deal with congestion situations, other collective behaviors such as stop-and-go waves or crowd turbulence lead to serious traffic perturbation that may have life-threatening effects [40]. Therefore, understanding the mechanisms underlying these collective behaviors would open the way for the design of bottom-up management strategies bound to promote smart collective behaviors and minimize the risks during mass events. Our results already suggest real-life applications to enhance traffic efficiency and walking comfort in crowded walkways. For instance, dividing the pavement into a “fast lane” and a “slow lane” would reduce the overall speed variability in the crowd, and therefore avoid the emergence of traffic breakdowns. This appears to be particularly suited to crowded pedestrian walkways in large cities, where local commuters often meet up with foreign tourists. In the future, insights about pedestrian crowds may also serve as a basis for the investigation of other kinds of crowds, such as groups of web users, traders at stock market, or consumers [21], [26], [30], [31].

Materials and Methods Experimental design Controlled experiments were conducted in May 2009 at the INRIA in Rennes, France. The goal was to observe the emergence of spontaneous traffic organization in bidirectional flows of walking pedestrians. A total of 119 participants took part in the study, which conformed to the Declaration of Helsinki. They were naïve to the purpose of our experiments, and gave written and informed consent to the experimental procedure. None had known pathology that would affect their locomotion. Experiments were conducted in a ring-shaped corridor with inner radius R in = 2 m and outer radius R out = 4.5 m, providing a total surface of 51.05 m2 (see Fig. 1) built in a larger experimental room. As a control experiment, each participant was first instructed to walk alone in the experimental corridor (see Fig. 6). Then, we studied the effect of pedestrian density on the emergence of collective patterns of motion. Experimental trials were made with N = 30, 50 and 60 pedestrians, corresponding to a global density level of 0.59, 0.98 and 1.18 p/m2, respectively. A total of 3, 2 and 6 replications were reconstructed and analyzed for N = 60, 50 and 30 participants, respectively. At the beginning of each trial, N participants were randomly distributed in the experimental corridor, and a walking direction was randomly attributed to each of them, in such a way that N/2 participants walked clockwise and N/2 anti-clockwise. At the starting signal, participants were asked to walk in their attributed direction as if they were moving alone in a street, and were not allowed to talk to each other (see Video S1). Each replication lasted for 60 seconds. The motion of each participant was recorded by means of an optoelectronic motion capture system (VICON MX-40, Oxford Metrics, UK). Participants were equipped with a white T-shirt and 4 reflexive markers, one on the forehead, one on the left acromion, and two on the right acromion to easily distinguish the left shoulder from right one. Markers motion was reconstructed using Vicon IQ software. The location of each participant was finally described as the center of mass of the 4 markers projected onto the horizontal plane (see Video S2). Clustering method Two pedestrians belong to the same cluster at a given moment of time if one of them is following the other. We assume that a pedestrian j is following another pedestrian i at time t, if the trajectory of j in the time segment passes at a distance smaller than from the location of pedestrian i at time t. This definition of the clustering method is illustrated in Fig. 2. The distance threshold was set to and the time window length was set to . In the supporting information it is shown that the parameter values do not significantly affect the clustering outcome, as long as these lie in a reasonable interval (see Text S1, Fig. S1 and 2). Simulation model Simulations were performed by means of the previously published heuristics-based model for pedestrian behavior [3]. The model describes the adaptation of the actual velocity of pedestrian i at time t by the acceleration equation , where is the relaxation time of 0.5 seconds, and the vector is the desired velocity pointing in direction and has the norm . The desired direction is given by minimizing the distance to the destination: where is the direction of the destination point Oi and the function is the distance to the first collision if pedestrian i moved in direction at his comfortable walking speed , taking into account the other pedestrians' walking speeds and body sizes. For simplicity, we represent the pedestrian's body by a circle of radius . If no collision is expected to occur in direction , is set to a default maximum value d max , which represents the “horizon distance” of pedestrian i. The direction is bounded by the vision field of the pedestrian, which ranges to the left and to the right by degrees with respect to the looking direction . The desired velocity is given by the equation , where d h is the distance between pedestrian i and the first obstacle in the desired direction at time t. In cases of overcrowding, physical interactions between bodies may occur, causing unintentional movements that are not determined by the above heuristics. Therefore, in situations where the pedestrian i would be in physical contacts with other pedestrians, a repulsive force is used instead , where g(x) is zero if the pedestrians i and j do not touch each other, and otherwise equals the argument x. is the normalized vector pointing from pedestrian j to i, and is the distance between the pedestrians' centers of mass. The physical interaction with a wall W is represented analogously by a contact force , where is the distance to the wall W and is the direction perpendicular to it. Here again, the contact force with walls vanishes when the pedestrian does not touch the wall. The resulting acceleration equation then reads and is solved together with the usual equation of motion , where denotes the location of pedestrian i at time t. In order to simulate the movement of a pedestrian turning in the ring-shaped corridor, the destination point Oi is updated at each simulation time step and located at a distance d O = 5 meters away in the direction tangent to the ring radius. The value of d O has been determined based on the control experiment results, by varying d O from 3 m to 10 m and choosing the value that minimizes the deviation between observed and predicted trajectories. The simulation parameters are = 0.5 s, = 45°, d max = 10 m, k = 103, R i = 0.2 m. Measurement functions The local density at time t and in direction is defined as the average value of the local density , for all points of the corridor located along the direction (with a reasonable spatial resolution). The local density is defined according to Ref. [40] as , where d jx is the distance between the center of mass of pedestrian j and location , and is a Gaussian-based weight function with R = 0.7 a weight parameter. The local radial speed is defined as the average radial speed of all pedestrians j located between directions and at time t, where the parameter is set to . The radial speed is given by , where r j is the radial position of pedestrian j in the experimental step.

Acknowledgments We are grateful to the M2S research group at the Université de Rennes 2 for their help and expertise during the experiments, and in particular to Armel Crétual, Richard Kulpa, Antoine Marin, and Anne-Hélène Olivier. We thank J.Gautrais, D.Helbing, J.Gouello and the two anonymous referees for inspiring comments on the work.

Author Contributions Conceived and designed the experiments: M. Moussaid, J. Pettré, C. Appert-Roland, P. Degond, G. Theraulaz. Performed the experiments: M. Moussaid, S. Lemercier, J. Pettré, C. Appert-Roland, G. Theraulaz. Analyzed the data: M. Moussaid, E.G. Guillot. Contributed reagents/materials/analysis tools: M. Moussaid, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron. Wrote the paper: M. Moussaid, J. Fehrenbach, O. Chabiron, J. Pettré, C. Appert-Roland, P. Degond, G. Theraulaz.