Universal Power Law Governing Pedestrian Interactions

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.

Figure 1

Analysis of anticipation effects in pedestrian motion. (a) Two pedestrians react strongly to avoid an upcoming collision even though they are far from each other (path segments over an interval of 4s are shown as colored lines, with arrows indicating acceleration). (b) In the same environment, two pedestrians walk close to each other without any relative acceleration. (c) The pair distribution function $g$ as a function of interpedestrian separation $r$ shows very different behavior when plotted for pedestrian pairs with different rate of approach $v=-dr/dt$. Units of $v$ are m/s. (d) In contrast, when $g$ is computed as a function of the time to collision $\tau$, curves corresponding to different $v$ collapse onto each other.

Figure 2

(a) The interaction energy computed from the dense Bottleneck data set and from the more sparse Outdoor data set (inset). The overall constant $k$ is normalized so that $E(1)=1$. Both data sets fit well to a power law up to a point marked $t_0$, beyond which there is no discernible interaction. Solid lines show the fit to the data and colored regions show their corresponding 95% confidence interval (Bottleneck, $R^2=0.94$; Outdoor, $R^2=0.92$). (b) The interaction energy in both data sets is well described by a power law with exponent 2.

Figure 3

Stills from simulations of agents following the force law derived from Eq. (2). In panels (a)–(d), agents are represented as cylinders and color coded according to their goal direction. The simulated agents display many emergent phenomena also seen in human crowds, including arching around narrow passages (a), clogging and “zipping” patterns at bottlenecks (c), and spontaneously self-organized lane formation (b) and (d). Panel (e) depicts a simulation of agents without a preferred goal direction (arrows represent the agents’ current orientations). The agents’ interactions lead to large-scale synchronization of their motion. Further simulation details are given in the Supplemental Material [29].

Figure 4

Inferred interaction energy $E\propto \ln (1/g)$ as a function of time to collision $\tau$ for different simulations, obtained using the anticipatory force described by Eq. (3), and the distance-dependent force described in Ref. [9] (inset). For simulations with strictly distance-dependent interactions, the inferred interaction energy does not show a dependence on $\tau$. In contrast, simulations following our model closely match the observed empirical power law for $E(\tau )$. Shaded regions denote average energy values $\pm 1$ standard deviation.