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For a group of pedestrians without any spatial boundaries, the methods of density estimation is a wide area of research. Besides, there is a specific difficulty when the density along one given pedestrian trajectory is needed in order to plot an 'individual-based' fundamental diagram. We illustrate why several methods become ill-defined in this case. We then turn to the widely used Voronoi-cell based density estimate. We show that for a typical situation of crossing flows of pedestrians, Voronoi method has to be adapted to the small sample size. We conclude with general remarks about the meaning of density measurements in such context. 

Patterns of crowd behavior are believed to result from local interactions between pedestrians. Many studies have investigated the local rules of interaction, such as steering, avoiding, and alignment, but how pedestrians control their walking speed when following another remains unsettled. Most pedestrian models assume the physical speed and distance of others as input. The present study compares such “omniscient” models with “visual” models based on optical variables.We experimentally tested eight speed control models from the pedestrian- and car-following literature. Walking participants were asked to follow a leader (a moving pole) in a virtual environment, while the leader’s speed was perturbed during the trial. In Experiment 1, the leader’s initial distance was varied. Each model was fit to the data and compared. The results showed that visual models based on optical expansion (θ˙) had the smallest root mean square error in speed across conditions, whereas other models exhibited increased error at longer distances. In Experiment 2, the leader’s size (pole diameter) was varied. A model based on the relative rate of expansion (θ˙/θ) performed better than the expansion rate model (θ˙), because it is less sensitive to leader size. Together, the results imply that pedestrians directly control their walking speed in one-dimensional following using relative rate of expansion, rather than the distal speed and distance of the leader. 

Conference Abstract 

While the cognitivist school of thought holds that the mind is analogous to a computer, performing logical operations over internal representations, the tradition of ecological psychology contends that organisms can directly ‘‘resonate’’ to information for action and perception without the need for a representational intermediary. The concept of resonance has played an important role in ecological psychology, but it remains a metaphor. Supplying a mechanistic account of resonance requires a non-representational account of central nervous system (CNS) dynamics. Towards this, we present a series of simple models in which a reservoir network with homeostatic nodes is used to control a simple agent embedded in an environment. This network spontaneously produces behaviors that are adaptive in each context, including (1) visually tracking a moving object, (2) substantially above-chance performance in the arcade game Pong, (2) and avoiding walls while controlling a mobile agent. Upon analyzing the dynamics of the networks, we find that behavioral stability can be maintained without the formation of stable or recurring patterns of network activity that could be identified as neural representations. These results may represent a useful step towards a mechanistic grounding of resonance and a view of the CNS that is compatible with ecological psychology. 

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.

 

When two streams of pedestrians cross at an angle, striped patterns spontaneously emerge as a result of local pedestrian interactions. This clear case of self-organized pattern formation remains to be elucidated. In counterflows, with a crossing angle of 180°, alternating lanes of traffic are commonly observed moving in opposite directions, whereas in crossing flows at an angle of 90°, diagonal stripes have been reported. Naka (1977) hypothesized that stripe orientation is perpendicular to the bisector of the crossing angle. However, studies of crossing flows at acute and obtuse angles remain underdeveloped. We tested the bisector hypothesis in experiments on small groups (18-19 participants each) crossing at seven angles (30° intervals), and analyzed the geometric properties of stripes. We present two novel computational methods for analyzing striped patterns in pedestrian data: (i) an edge-cutting algorithm, which detects the dynamic formation of stripes and allows us to measure local properties of individual stripes; and (ii) a pattern-matching technique, based on the Gabor function, which allows us to estimate global properties (orientation and wavelength) of the striped pattern at a time T . We find an invariant property: stripes in the two groups are parallel and perpendicular to the bisector at all crossing angles. In contrast, other properties depend on the crossing angle: stripe spacing (wavelength), stripe size (number of pedestrians per stripe), and crossing time all decrease as the crossing angle increases from 30° to 180°, whereas the number of stripes increases with crossing angle. We also observe that the width of individual stripes is dynamically squeezed as the two groups cross each other. The findings thus support the bisector hypothesis at a wide range of crossing angles, although the theoretical reasons for this invariant remain unclear. The present results provide empirical constraints on theoretical studies and computational models of crossing flows. 

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.

 

Patterns of collective motion in bird flocks, fish schools and human crowds are believed to emerge from local interactions between individuals. Most ‘flocking' models attribute these local interactions to hypothetical rules or metaphorical forces and assume an omniscient third-person view of the positions and velocities of all individuals in space. We develop a visual model of collective motion in human crowds based on the visual coupling that governs pedestrian interactions from a first-person embedded viewpoint. Specifically, humans control their walking speed and direction by cancelling the average angular velocity and optical expansion/contraction of their neighbours, weighted by visibility (1 − occlusion). We test the model by simulating data from experiments with virtual crowds and real human ‘swarms'. The visual model outperforms our previous omniscient model and explains basic properties of interaction: ‘repulsion' forces reduce to cancelling optical expansion, ‘attraction' forces to cancelling optical contraction and ‘alignment' to cancelling the combination of expansion/contraction and angular velocity. Moreover, the neighbourhood of interaction follows from Euclid's Law of perspective and the geometry of occlusion. We conclude that the local interactions underlying human flocking are a natural consequence of the laws of optics. Similar perceptual principles may apply to collective motion in other species. 

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.