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1. A visual model of collective motion in human crowds

   Warren, W.H. ; Dachner, G. ( February 2020 , Perception)

   null (Ed.)

   Collective motion in human crowds emerges from local interactions between individual pedestrians. Previously, we found that an individual in a crowd aligns their velocity vector with a weighted average of their neighbors’ velocities, where the weight decays with distance (Rio, Dachner, & Warren, PRSB, 2018; Warren, CDPS, 2018). Here, we explain this “alignment rule” based solely on visual information. When following behind a neighbor, the follower controls speed by canceling the neighbor’s optical expansion (Bai & Warren, VSS, 2018) and heading by canceling the neighbor’s angular velocity. When walking beside a neighbor, these relations reverse: Speed is controlled by canceling angular velocity and heading by canceling optical expansion. These two variables trade off as sinusoidal functions of eccentricity (Dachner & Warren, VSS, 2018). We use this visual model to simulate the trajectories of participants walking in virtual (12 neighbors) and real (20 neighbors) crowds. The model accounts for the data with root mean square errors (.04–.05 m/s, 1.5°–2.0°) the distance decay as a consequence of comparable to those of our previous velocity-alignment model. Moreover, the model explains Euclid’s law of perspective, without an explicit distance term. The visual model thus provides a better explanation of collective motion. 

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2. Local interactions underlying collective motion in human crowds

   https://doi.org/10.1098/rspb.2018.0611  

   Rio, Kevin W. ; Dachner, Gregory C. ; Warren, William H. ( May 2018 , Proceedings of the Royal Society B: Biological Sciences)

   null (Ed.)

   It is commonly believed that global patterns of motion in flocks, schools and crowds emerge from local interactions between individuals, through a process of self-organization. The key to explaining such collective behaviour thus lies in deciphering these local interactions. We take an experiment-driven approach to modelling collective motion in human crowds. Previously, we observed that a pedestrian aligns their velocity vector (speed and heading direction) with that of a neighbour. Here we investigate the neighbourhood of interaction in a crowd: which neighbours influence a pedestrian's behaviour, how this depends on neighbour position, and how the influences of multiple neighbours are combined. In three experiments, a participant walked in a virtual crowd whose speed and heading were manipulated. We find that neighbour influence is linearly combined and decreases with distance, but not with lateral position (eccentricity). We model the neighbourhood as (i) a circularly symmetric region with (ii) a weighted average of neighbours, (iii) a uni-directional influence, and (iv) weights that decay exponentially to zero by 5 m. The model reproduces the experimental data and predicts individual trajectories in observational data on a human ‘swarm’. The results yield the first bottom-up model of collective crowd motion. 

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3. Is the neighborhood of interaction in human crowds metric, topological, or visual?

   https://doi.org/10.1093/pnasnexus/pgad118  

   Wirth, Trenton D ; Dachner, Gregory C ; Rio, Kevin W ; Warren, William H ( May 2023 , PNAS Nexus)

   Borge-Holthoefer, Javier (Ed.)

   Global patterns of collective motion in bird flocks, fish schools, and human crowds are thought to emerge from local interactions within a neighborhood of interaction, the zone in which an individual is influenced by their neighbors. Both metric and topological neighborhoods have been reported in animal groups, but this question has not been addressed for human crowds. The answer has important implications for modeling crowd behavior and predicting crowd disasters such as jams, crushes, and stampedes. In a metric neighborhood, an individual is influenced by all neighbors within a fixed radius, whereas in a topological neighborhood, an individual is influenced by a fixed number of nearest neighbors, regardless of their physical distance. A recently proposed alternative is a visual neighborhood, in which an individual is influenced by the optical motions of all visible neighbors. We test these hypotheses experimentally by asking participants to walk in real and virtual crowds and manipulating the crowd's density. Our results rule out a topological neighborhood, are approximated by a metric neighborhood, but are best explained by a visual neighborhood that has elements of both. We conclude that the neighborhood of interaction in human crowds follows naturally from the laws of optics and suggest that previously observed “topological” and “metric” interactions might be a consequence of the visual neighborhood.

    

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4. Human Crowds as Social Networks: Collective Dynamics of Consensus and Polarization

   https://doi.org/10.1177/17456916231186406  

   Warren, William H. ; Falandays, J. Benjamin ; Yoshida, Kei ; Wirth, Trenton D. ; Free, Brian A. ( August 2023 , Perspectives on Psychological Science)

   A ubiquitous type of collective behavior and decision-making is the coordinated motion of bird flocks, fish schools, and human crowds. Collective decisions to move in the same direction, turn right or left, or split into subgroups arise in a self-organized fashion from local interactions between individuals without central plans or designated leaders. Strikingly similar phenomena of consensus (collective motion), clustering (subgroup formation), and bipolarization (splitting into extreme groups) are also observed in opinion formation. As we developed models of crowd dynamics and analyzed crowd networks, we found ourselves going down the same path as models of opinion dynamics in social networks. In this article, we draw out the parallels between human crowds and social networks. We show that models of crowd dynamics and opinion dynamics have a similar mathematical form and generate analogous phenomena in multiagent simulations. We suggest that they can be unified by a common collective dynamics, which may be extended to other psychological collectives. Models of collective dynamics thus offer a means to account for collective behavior and collective decisions without appealing to a priori mental structures.

    

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5. Robust Weighted Averaging Accounts for Recruitment Into Collective Motion in Human Crowds

   https://doi.org/10.3389/fams.2021.761445  

   Wirth, Trenton D. ; Warren, William H. ( November 2021 , Frontiers in Applied Mathematics and Statistics)

   Agent-based models of “flocking” and “schooling” have shown that a weighted average of neighbor velocities, with weights that decay gradually with distance, yields emergent collective motion. Weighted averaging thus offers a potential mechanism of self-organization that recruits an increasing, but self-limiting, number of individuals into collective motion. Previously, we identified and modeled such a ‘soft metric’ neighborhood of interaction in human crowds that decays exponentially to zero at a distance of 4–5 m. Here we investigate the limits of weighted averaging in humans and find that it is surprisingly robust: pedestrians align with the mean heading direction in their neighborhood, despite high levels of noise and diverging motions in the crowd, as predicted by the model. In three Virtual Reality experiments, participants were immersed in a crowd of virtual humans in a mobile head-mounted display and were instructed to walk with the crowd. By perturbing the heading (walking direction) of virtual neighbors and measuring the participant’s trajectory, we probed the limits of weighted averaging. 1) In the “Noisy Neighbors” experiment, the neighbor headings were randomized (range 0–90°) about the crowd’s mean direction (±10° or ±20°, left or right); 2) in the “Splitting Crowd” experiment, the crowd split into two groups (heading difference = 10–40°) and the proportion of the crowd in one group was varied (50–84%); 3) in the “Coherent Subgroup” experiment, a perturbed subgroup varied in its coherence (heading SD = 0–20°) about a mean direction (±10° or ±20°) within a noisy crowd (heading range = 180°), and the proportion of the crowd in the subgroup was varied. In each scenario, the results were predicted by the weighted averaging model, and attraction strength (turning rate) increased with the participant’s deviation from the mean heading direction, not with group coherence. However, the results indicate that humans ignore highly discrepant headings (45–90°). These findings reveal that weighted averaging in humans is highly robust and generates a common heading direction that acts as a positive feedback to recruit more individuals into collective motion, in a self-reinforcing cascade. Therefore, this “soft” metric neighborhood serves as a mechanism of self-organization in human crowds. 

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