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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. 

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. 

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.