Collective Motion in Human Crowds

Abstract

The balletic motion of bird flocks, fish schools, and human crowds is believed to emerge from local interactions between individuals in a process of self-organization. The key to explaining such collective behavior thus lies in understanding these local interactions. After decades of theoretical modeling, experiments using virtual crowds and analysis of real crowd data are enabling us to decipher the “rules of engagement” governing these interactions. On the basis of such results, my students and I built a dynamical model of how a pedestrian aligns his or her motion with that of a neighbor and how these binary interactions are combined within a neighborhood of interaction. Computer simulations of the model generate coherent motion at the global level and reproduce individual trajectories at the local level. This approach has yielded the first experiment-driven, bottom-up model of collective motion, providing a basis for understanding more complex patterns of crowd behavior in both everyday and emergency situations.

Keywords

crowd behavior pedestrian dynamics flocking collective behavior self-organization

The spectacle of a murmuration of starlings careening in near-perfect synchrony, or a school of herring smoothly circling in a “mill” formation, are prime examples of collective behavior in biological systems. Humans exhibit such collective motion as well, as when a crowd of pedestrians adopts a common motion on the way to a train platform or forms opposing lanes of traffic in a shopping mall. Collective behavior can also go tragically awry, such as in a stampede in a stadium or a rush to one exit in a burning nightclub. How do many individuals spontaneously coalesce into a coherently moving body? After decades of theoretical modeling, research has now advanced to the point that experimentally grounded models of collective motion are possible.

The Self-Organization of Behavior

Beyond the intrinsic fascination that collective motion holds, it is a paradigmatic case of self-organized behavior. There is a growing recognition among biologists, psychologists, and cognitive scientists that general principles of self-organization hold promise for explaining the organization of human and animal behavior at multiple levels, without needing to appeal to a priori neural or cognitive structures (Camazine et al., 2001; Goldstone & Gureckis, 2009; Kelso, 1995).

The ingredients of self-organization have been identified in systems ranging from lasers to ant trails (Haken, 1983). These ingredients consist of an open system that dissipates energy and is composed of many interacting components, which are locally coupled by physical or information fields; fluctuation that nudges the system away from disorder;¹ and positive feedback that amplifies this initial fluctuation, capturing more components to yield an emergent spatial or temporal pattern.

One can see this process at work in fish schooling. Foraging fish (dissipative components) in a shoal are locally coupled by sensory information, but each fish is randomly oriented (disorder). The approach of a predator triggers nearby fish to escape by swimming in a similar direction (fluctuation), which progressively recruits neighboring fish into the emerging pattern (positive feedback) to form a coherently moving school. Collective motion has become a test bed for self-organized behavior because these mechanisms are observable and amenable to experimental study.

Modeling Collective Motion

If collective behavior emerges from local interactions between individuals, then the crux of the problem is understanding the “rules of engagement” that govern these interactions. This insight has motivated a raft of microscopic or agent-based models focused on the local level of individual behavior. They are complementary to macroscopic models, which treat a huge crowd as a fluid and focus on global properties such as density and mean velocity (Cristiani, Piccoli, & Tosin, 2014).

The dominant class of microscopic models is the attraction-repulsion framework, which was derived from early research on fish schooling (Schellinck & White, 2011). It assumes three basic rules (Fig. 1): (a) attraction—move toward neighbors in a far zone, (b) repulsion—move away from neighbors in a near zone, and (c) alignment—match the velocity (speed and heading direction) of neighbors in an intermediate zone. By adjusting the radii of these zones, the model can generate unaligned aggregation (shoaling), strongly aligned translation (schooling), and rotational motion (mills; Couzin, Krause, James, Ruxton, & Franks, 2002). Reynolds (1987) famously applied this model to computer animation, producing the wildebeest stampede in The Lion King and the bat swarms in Batman.

Fig. 1.

The attraction-repulsion framework. An individual is repelled from neighbors in the near zone (orange), aligns his or her velocity with neighbors in the intermediate zone (green), and is attracted to neighbors in the far zone (blue). Figure reprinted from Giardina (2008; copyright Taylor & Francis Group, www.tandfonline.com).

The stripped-down self-propelled-particle model (Czirók & Vicsek, 2000) subsequently showed that a velocity-based alignment rule alone can generate translational motion, as well as transitions from incoherent to coherent motion. Conversely, the social-force model (Helbing & Molnár, 1995) was predicated on position-based attraction and repulsion rules. It, too, generates plausible patterns of global motion, but individual trajectories do not resemble human locomotion at the local level (Pelechano, Allbeck, & Badler, 2007). More recently, vision-based models proposed that local interactions are driven by visual or cognitive heuristics (Moussaïd, Helbing, & Theraulaz, 2011; Ondrˇej, Pettré, Olivier, & Donikian, 2010).

With recent advances in 3-D motion tracking of animals and humans, researchers are starting to compare such theoretical models with observational data, yielding many tantalizing insights (Cavagna et al., 2010; Charalambous, Karamouzas, Guy, & Chrysanthou, 2014; Hildenbrandt, Carere, & Hemelrijk, 2010; Lukeman, Li, & Edelstein-Keshet, 2010; Moussaïd et al., 2012; Wolinski et al., 2014; Zhang, Klingsch, Schadschneider, & Seyfried, 2012). Computer simulations of observational data are, however, insufficient to test local rules. It has been recognized that the same motion pattern can be generated by different local rules (Vicsek & Zafeiris, 2012; Weitz et al., 2012), an example of degeneracy in complex systems. Deciphering the rules and the perceptual information that actually govern local interactions thus requires experimental manipulation of individual behavior at the local level (Gautrais et al., 2012; Sumpter, Mann, & Perna, 2012).

Behavioral Dynamics

My students and I are pursuing just such an experiment-driven, bottom-up approach to human crowd behavior (for a similar approach in fish, see Gautrais et al., 2012; Zienkiewicz, Barton, Porfiri, & Di Bernardo, 2015). In the behavioral dynamics approach (Warren, 2006), we first study each basic behavior and model it as a dynamical system. Related experiments determine the visual information that controls that behavior (Gibson, 1979; Warren, 1998). The resulting control laws are analogous to local “rules” but emphasize their continuous dynamical rather than logical form.

Building a pedestrian model

We began by building a pedestrian model that captures how people walk through an environment of goals, obstacles, and moving objects (Fajen & Warren, 2003; Warren & Fajen, 2008). We tested participants walking in virtual reality (VR), making it possible to manipulate the visual environment, and modeled their locomotor trajectories. To get an intuition for the model, imagine that a pedestrian’s heading direction is attached to the goal direction by a damped spring (Fig. 2a). As the pedestrian walks forward, his or her heading is pulled into alignment with the goal (an attractor); conversely, another spring pushes the pedestrian’s heading away from the direction of an obstacle (a repeller). The path of locomotion is the result of all such “spring forces” acting on the pedestrian as he or she moves through the environment. This simple attractor/repeller dynamic successfully models the emergence of locomotor paths and predicts paths in more complex environments.

Fig. 2.

Pedestrian model and velocity-alignment dynamics. In the pedestrian model (a), as a pedestrian walks with speed s, his or her heading direction is attracted to the goal direction by a damped spring with negative stiffness (–k_g) and repelled from the obstacle direction by a damped spring with positive stiffness (+k_o); stiffness decays exponentially with distance (d_g and d_o). The current heading is the result of all spring forces, which evolve over time, yielding an emergent locomotor path. In the alignment-dynamics model, a follower is attracted to either (b) a leader’s speed by a spring with stiffness k_s or (c) a leader’s heading direction by spring with stiffness –k_h.

Alignment dynamics

To scale up to crowds, we next studied binary interactions between two pedestrians—in particular, how one person follows another (Dachner & Warren, 2014; Rio, Rhea, & Warren, 2014). We found that rather than keeping a constant distance from the leader, the follower matches the leader’s velocity—similar to an alignment rule. The results allowed us to formulate a simple model of alignment dynamics, in which the follower linearly accelerates to match the leader’s speed (Fig. 2b) and angularly accelerates to match the leader’s heading (Fig. 2c). By manipulating the visual information for the leader in VR, we have determined that the follower does this by jointly canceling the leader’s optical expansion and angular velocity, yielding a visual control law for following that decreases with distance (Dachner & Warren, 2017). We believe these alignment dynamics form the basis of collective motion.

A Pedestrian’s Neighborhood

In a crowd, however, each pedestrian is influenced by multiple neighbors, so the key to collective motion is a pedestrian’s zone of influence, or neighborhood of interaction. We thus set out to determine how the influences of multiple neighbors are combined within a neighborhood. To probe these interactions experimentally, we created a novel VR paradigm in which a participant is asked to walk with a virtual crowd, allowing us to manipulate the speed and heading of neighbors and measure the participant’s response. We compared the results with motion-capture data on a human “swarm,” in which a group of 16 to 20 participants were asked to walk about a large hall, veering randomly left and right, while staying together. What we discovered enabled us to formulate the first bottom-up model of collective motion.

Superposition and coupling strength

Nearly all models of collective motion assume that binary interactions between a pedestrian and each neighbor are linearly combined, a principle known as superposition. To test this basic assumption, we perturbed the speed or heading of a subset of the 12 virtual neighbors (Fig. 3a) and measured the participant’s change in speed or heading (Rio, Dachner, & Warren, 2018). We found that the mean response increased linearly with the number of perturbed neighbors, supporting superposition (the correlation coefficient, a measure of agreement that varies between 0 and 1, was .99). This proportional response was observed on individual trials.

Fig. 3.

Testing the responses of a participant walking with a virtual crowd of 12 neighbors. The heading direction (or speed) of a subset of neighbors in a near zone or a far zone is perturbed midway through a 12-s trial (a). The perturbed subset (S) could contain 0, 3, 6, 9, or 12 neighbors; this example shows 3, highlighted in red. The graph (b) shows that participants’ mean final heading in the last 2 s (solid curves) increased with the number of perturbed neighbors, but the response was greater to near (blue) than far (red) neighbors. Simulations of the crowd model (dotted curves) were nearly identical, within the 95% confidence interval of the human data (shaded regions). Figure modified from Rio, Dachner, and Warren (2018).

However, we also found that the response to near neighbors (~1.8 m) was significantly greater than to far neighbors (~3.5 m; Fig. 3b). Most zonal models assume a constant coupling strength with a hard radius (Fig. 1), but analysis of the human swarm data (Fig. 4) confirmed that coupling strength decreases exponentially with distance, going to zero at 4 to 5 m. Such a “soft” radius is advantageous for collective motion (Cucker & Smale, 2007). We thus modeled the neighborhood as a weighted average of neighbors, where the weight decays exponentially with distance. This might have a visual basis, for a neighbor’s visual angle and angular velocity decrease with distance because of the laws of perspective, and farther neighbors are progressively occluded by nearer neighbors in a crowd.

Fig. 4.

Data from the human swarm: heat map of mean absolute heading difference between the participant nearest the center of the swarm (at [0, 0], heading upward) and each neighbor, averaged over 6 min of data (three 2-min trials, initial interpersonal distance of 2 m; cell = 0.5 m × 0.5 m). Reprinted from Rio, Dachner, and Warren (2018).

On the other hand, coupling strength did not consistently depend on a neighbor’s eccentricity (lateral position). This result suggests that the human neighborhood is circular, which is borne out by the swarm data (Fig. 4). Yet coupling strength obviously drops to zero at the edges of the field of view. Prey species often have nearly panoramic vision, implying that flocks and schools are bidirectionally coupled (but see Nagy, Akros, Biro, & Vicsek, 2010). In contrast, humans have an approximately 180° visual field, which implies that crowds are unidirectionally coupled, so pedestrians respond only to neighbors in front of them. This suspicion is confirmed by analysis of time delays in the swarm data, which shows that a pedestrian turns after neighbors ahead, followed by neighbors behind. The observation has important implications for how crowds steer and make collective decisions.

Metric or topological?

Most zonal models assume that the neighborhood of interaction (Fig. 5) is defined by metric distance, such that an individual is influenced by all neighbors within a fixed radius (e.g., 4 m). In contrast, evidence suggests that starlings have neighborhoods defined by topological distance, or a fixed number of nearest neighbors (e.g., seven) regardless of their absolute distance (Ballerini et al., 2008; but see Evangelista, Ray, Raja, & Hedrick, 2017). This adaptation would ensure cohesive flocking despite large variation in the density of a swooping murmuration and prevent birds drifting away from the flock.

Fig. 5.

Metric and topological neighborhoods. According to the metric hypothesis, the participant is influenced by all neighbors within a fixed radius (shaded region); according to the topological hypothesis, the participant is influenced by a fixed number of nearest neighbors (dotted red lines). In the high-density condition (a), both metric and topological neighborhoods contain perturbed (red) and unperturbed (black) neighbors. In the low-density condition (b), the metric neighborhood contains fewer unperturbed neighbors (gray) than before, so the metric hypothesis predicts a greater response to the perturbed neighbors (red). But the nearest neighbors remain the same, so the topological hypothesis predicts the same response to the perturbed neighbors.

We tested these hypotheses in humans by manipulating the density of a virtual crowd of 12 neighbors. In the critical experiment (Wirth & Warren, 2016), we perturbed the heading of the nearest 2 to 4 neighbors, who always appeared at constant distances, while unperturbed neighbors varied in distance (Fig. 5). The topological hypothesis predicts that density should have had no effect on the response. In contrast, the metric hypothesis predicts that responses should have been greater in the low-density condition, when fewer unperturbed neighbors appeared in the neighborhood, than in the high-density condition. That is precisely what we found, consistent with the metric hypothesis.

The donut model

However, it seemed unlikely that a group of neighbors outside a metric radius of 4 m would be completely ignored. To check, we varied the distance of the entire virtual crowd up to 8 m—and found that the participant still responded to heading perturbations. The response decreased with distance, but at a much more gradual rate than before. This result raised the prospect of a flexible neighborhood shaped like a donut, with two decay rates (see Fig. 6): a slow decay to the nearest neighbor in the crowd (the donut hole) and a faster decay within the crowd (the ring of the donut).

Fig. 6.

The donut model of a pedestrian’s neighborhood. The pedestrian’s response is a weighted average of i neighbors, where the weight decays gradually with distance to the nearest neighbor (d_nn) up to 11 m, and more rapidly within the crowd (d_i – d_nn = r = 4–5 m), creating a flexible neighborhood with a soft radius.

To test the donut hypothesis, we varied the distance of the whole crowd and selectively perturbed the near, middle, or far row of neighbors. As expected, we observed a gradual decay to the nearest neighbor, which went to zero at about 11 m, followed by a more rapid decay within the crowd, which went to zero 4 to 5 m from the nearest neighbor. Such a flexible neighborhood with a variable-size donut hole would allow pedestrians to coordinate their motion with a group up to 11 m away, so they do not drift away from the “flock.”

Taken together, the experimental results enabled us to formalize the donut model of a pedestrian’s neighborhood (Fig. 6) as circularly symmetric, with a unidirectional coupling to neighbors within ±90° of the heading direction. The influences of these neighbors are combined as a weighted average, with weights that decay exponentially with metric distance at two decay rates.

Crowd Dynamics

Combining the neighborhood model (Fig. 6) with the alignment dynamics (Fig. 2) gives us a model of the local rule of engagement underlying collective motion. Specifically, a pedestrian’s linear (or angular) acceleration is a weighted average of the difference between his or her current speed (or heading) and that of each neighbor, where the weight decays exponentially with distance. We tested this model against human data at the global and local levels.

Global motion

First, to demonstrate that the model generates globally coherent motion, we performed multiagent simulations of 30 freely interacting agents. The model agents were assigned initial positions and random initial headings and speeds and then let go, simulating the trajectories for 30 s. The model converged to collective motion for a wide range of initial headings (80°) and initial speeds (0.8 m/s) with a variability comparable with that of the human swarm. The donut model converged over a wider range of crowd densities than a simpler disk model without the donut hole. The model thus reproduces the basic phenomenon of common motion with a coherence comparable with that of human crowds.

Local trajectories

We cannot expect to simulate individual trajectories this way, however—that would be akin to predicting the motions of individual molecules in a gas. Instead, we simulated participants in the swarm one at a time, treating the positions and velocities of their neighbors as input (Fig. 7; Rio et al., 2018). We simulated thirty-one 10-s segments of swarm data using the simple disk model, as neighbors were close together. The mean correlation (r) between model and human trajectories was .92 for heading and .62 for speed (a bit lower because there was little speed variation in the swarm).

Fig. 7.

Simulation of a sample trajectory from the human swarm: (a) path in space (dots at 1-s intervals), (b) time series of speed, and (c) time series of heading. The solid red curve corresponds to a participant, the dashed blue curve to a model agent, and the black curves to neighbors that were input to the model. Simulations were performed on thirty-one 10-s segments of swarm data that had continuous tracking of seven or more neighbors. Reprinted from Rio, Dachner, and Warren (2018).

We also simulated the trajectories of participants walking in the virtual crowd experiment, when the neighbors’ speed or heading was perturbed. The model closely reproduced human trajectories (mean r = .88 for changes in heading, mean r = .90 for changes in speed). Moreover, the model’s final heading and speed were virtually identical with the mean human data in each condition (Fig. 3). The model thus reproduces pedestrian motion at the local as well as the global level.

The neighborhood as a mechanism of self-organization

Essential to self-organization is a positive feedback mechanism that progressively recruits more individuals into the emerging pattern. In collective motion, this role is played by the local neighborhood. Each individual is visually coupled to multiple neighbors ahead and influences others behind. As neighbors begin to align, they increasingly influence the neighborhoods that contain them, drawing more individuals into alignment. In this manner, a pattern of coherent motion propagates through the crowd.

Conclusion

Thanks to experiments with virtual crowds and motion-capture data on real crowds, we are beginning to decipher the local rules that underlie collective crowd behavior. This approach has yielded the first experiment-driven, bottom-up model of collective motion, accounting for both globally coherent motion and local trajectories. Challenges ahead include building an information-based model with visual control laws, explaining more complex patterns of crowd behavior, formally linking the microscopic and macroscopic levels of description (Cristiani et al., 2014; Degond, Appert-Rolland, Moussaïd, Pettré, & Theraulaz, 2013), and understanding the causes of crowd disasters to improve evacuation planning (Helbing, Buzna, Johansson, & Werner, 2005; Schadschneider, Klingsch, Klüpfel, Rogsch, & Seyfried, 2009).

Footnotes

Acknowledgements

The author would like to thank Kevin Rio, Greg Dachner, Trent Wirth, Stéphane Bonneaud, Emily Richmond, Arturo Cardenas, Michael Fitzgerald, and the team who helped collect and process data from the Sayles Swarm.

Action Editor

Randall W. Engle served as action editor for this article.

Declaration of Conflicting Interests

The author(s) declared that there were no conflicts of interest with respect to the authorship or the publication of this article.

Funding

This research was supported by National Institutes of Health Grant R01EY010923 and National Science Foundation Grant BCS-1431406.

Notes

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Collective Motion in Human Crowds

Abstract

Keywords

The Self-Organization of Behavior

Modeling Collective Motion

Behavioral Dynamics

Building a pedestrian model

Alignment dynamics

A Pedestrian’s Neighborhood

Superposition and coupling strength

Metric or topological?

The donut model

Crowd Dynamics

Global motion

Local trajectories

The neighborhood as a mechanism of self-organization

Conclusion

Recommended Reading

Footnotes

Acknowledgements

Action Editor

Declaration of Conflicting Interests

Funding

Notes

References