Velocity correlations and spatial dependencies between neighbors in a unidirectional flow of pedestrians

The aim of the paper is an analysis of self-organization patterns observed in the unidirectional flow of pedestrians. On the basis of experimental data from Zhang et al. [J. Zhang et al., J. Stat. Mech. (2011) P06004], we analyze the mutual positions and velocity correlations between pedestrians when walking along a corridor. The angular and spatial dependencies of the mutual positions reveal a spatial structure that remains stable during the crowd motion. This structure differs depending on the value of $n$, for the consecutive $n\text{th}$-nearest-neighbor position set. The preferred position for the first-nearest neighbor is on the side of the pedestrian, while for further neighbors, this preference shifts to the axis of movement. The velocity correlations vary with the angle formed by the pair of neighboring pedestrians and the direction of motion and with the time delay between pedestrians' movements. The delay dependence of the correlations shows characteristic oscillations, produced by the velocity oscillations when striding; however, a filtering of the main frequency of individual striding out reduces the oscillations only partially. We conclude that pedestrians select their path directions so as to evade the necessity of continuously adjusting their speed to their neighbors'. They try to keep a given distance, but follow the person in front of them, as well as accepting and observing pedestrians on their sides. Additionally, we show an empirical example that illustrates the shape of a pedestrian's personal space during movement.

Figure 1

Concept of the $n\text{th}$-nearest neighbor and angle between neighbors in a unidirectional flow. For the central pedestrian (orange), the pedestrian on the right is the first-nearest neighbor. Consecutive neighbors are marked on the figure. Here $\mathrm{\Theta }$ denotes the angle between the orange pedestrian and the first-nearest neighbor.

Figure 2

Histogram of distances for the $n\text{th}$-nearest neighbor.

Figure 3

Spatial distribution of the (a) first-, (b) second-, (c) third-, (d) fourth-, (e) fifth-, and (f) sixth-nearest neighbors around moving pedestrians in the analyzed experiment. The direction of motion corresponds to the top of the plot. Each point on the pixel corresponds to 1 ${\text{cm}}^2$. The redder the color is the more often the $n\text{th}$-nearest neighbor was detected at this particular position.

Figure 4

Sum of the spatial distribution histogram for four nearest neighbors $n\in \langle 1,4\rangle$. The black ellipse in the center represents the typical body size [15]. The central white area is forbidden for other pedestrians.

Figure 5

Histogram of angular distribution of the (a) first-, (b) second-, (c) third-, (d) fourth-, (e) fifth-, and (f) sixth-nearest neighbors according to $\mathrm{\Theta }$ for $n\le 6$. A zero angle (top) corresponds to the direction of motion. Angular discretization is $1^{\circ }$. The radius of the plot corresponds to 400 detected positions of the $n\text{th}$ neighbor at a given $\mathrm{\Theta }$. Assuming approximately $79800$ total $n\text{th}$ neighbors detected for each $n$ corresponds to a $0.5\%$ chance of finding the $n\text{th}$ neighbor at this $\mathrm{\Theta }$.

Figure 6

Time dependence of the velocity component parallel to the corridor, for two neighboring pedestrians.

Figure 7

Fourier spectrum of the velocity of an exemplary pedestrian.

Figure 8

Angle- and delay-dependent velocity correlations between (a) first-, (b) second-, and (c) third-nearest neighbors within a given $\mathrm{\Theta }$ and (d) first-, (e) second-, and (f) third-nearest neighbors within a given $\mathrm{\Theta }$ after the frequency of striding is filtered out.