Human crowds often bear a striking resemblance to interacting particle systems, and this has prompted many researchers to describe pedestrian dynamics in terms of interaction forces and potential energies. The correct quantitative form of this interaction, however, has remained an open question. Here, we introduce a novel statistical-mechanical approach to directly measure the interaction energy between pedestrians. This analysis, when applied to a large collection of human motion data, reveals a simple power law interaction that is based not on the physical separation between pedestrians but on their projected time to a potential future collision, and is therefore fundamentally anticipatory in nature. Remarkably, this simple law is able to describe human interactions across a wide variety of situations, speeds and densities. We further show, through simulations, that the interaction law we identify is sufficient to reproduce many known crowd phenomena. In terms of its large-scale behaviors, a crowd of pedestrians can look strikingly similar to many other collections of repulsively-interacting particles [1][2][3][4]. These similarities have inspired a variety of pedestrian crowd models, including cellular automata and continuum-based approaches [5][6][7][8], as well as simple particle or agent-based models [9][10][11][12][13][14][15]. Many of these models conform to a longstanding hypothesis that humans in a crowd interact with their neighbors through some form of "social potential" [16], analogous to the repulsive potential energies between physical particles. How to best determine the quantitative form of this interaction potential, however, has remained an open question, with most previous researchers employing a simulation-driven approach. Previously, direct measurement of the interaction law between pedestrians has been confounded by two primary factors. First, each individual in a crowd experiences a complex environment of competing forces, making it difficult to isolate and robustly quantify a single pairwise interaction. Secondly, a pedestrian's motion is strongly influenced not just by the present position of neighboring pedestrians, but by their anticipated future positions [17] [18] [19] [20] [21] , a fact which has influenced recent models [22] [23] [24] [25] . Consider, for example, two well-separated pedestrians walking into a head-on collision (Fig. 1a) . These pedestrians typically exhibit relatively large acceleration as they move to avoid each other, as would result from a large repulsive force. On the other hand, pedestrians walking in parallel directions exhibit almost no acceleration, even when their mutual separation is small (Fig. 1b) . Here, we address both of the aforementioned factors using a data-driven, statistical mechanics-based analysis that accounts properly for the anticipatory nature of human interactions. This approach allows us to directly and robustly measure the interaction energy between pedestrians. The consistency of our measurements across a variety of settings suggests a simple and universal law governing pedestrian motion. To perform our analysis, we turn to the large collections of recently published crowd datasets recorded by motion-capture or computer vision-based techniques. These datasets include pedestrian trajectories from several outdoor environments [26, 27] and controlled lab settings [28] (a summary of datasets is given in the Supplemental Material [29] ). To reduce statistical noise, datasets with similar densities were combined together, resulting in one Outdoor dataset comprising 1,146 trajectories of pedestrians in sparse-to-moderate outdoor settings, and one Bottleneck dataset with 354 trajectories of pedestrians in dense crowds passing through narrow bottlenecks. In analyzing these datasets, our primary tool for quantifying the strength of interactions between pedestrians is the statistical-mechanical pair distribution function, denoted g. As in the typical condensed matter setting [30], here we define the pair distribution function g(x) as the observed probability density for two pedestrians to have relative separation x divided by the expected probability density for two non-interacting pedestrians to have the same separation. In general, the probability density for noninteracting pedestrians cannot be known a priori, since it depends on where and how frequently pedestrians enter and exit the environment. However, for large datasets we are able to closely approximate this distribution by sampling the separation between all pairs of pedestrians that are not simultaneously present in the scene (and therefore not interacting). As defined above, small values of the pair distribution function, g(x) 1, correspond to situations where interactions produce strong avoidance. If the Cartesian distance r between two pedestrians were a sufficient descriptor of their interaction, we would expect the shape of the pair distribution function g(r) to be independent of all other variables. However, as can be seen in Fig. 1c, g(r) has large, qualitative differences when the data is binned by the rate at which the two pedestrians are approaching each other, v = −dr/dt. In particular, pedestrians with a small rate of approach are more likely to be found close together than those that are
An interaction energy between pedestrians was modeled with a power law in recent and concurrent work by REF .
Universal power law governing pedestrian interactions
Phys. Rev. Lett. 113, 238701 (2014)
Physics Medicine