Spatial-size scaling of pedestrian groups under growing density conditions

We study the dependence on crowd density of the spatial size, configuration, and velocity of pedestrian social groups. We find that, in the investigated density range, the extension of pedestrian groups in the direction orthogonal to that of motion decreases linearly with the pedestrian density around them, both for two- and three-person groups. Furthermore, we observe that at all densities, three-person groups walk slower than two-person groups, and the latter are slower than individual pedestrians, the differences in velocities being weakly affected by density. Finally, we observe that three-person groups walk in a V-shaped formation regardless of density, with a distance between the pedestrians in the front and back again almost independent of density, although the configuration appears to be less stable at higher densities. These findings may facilitate the development of more realistic crowd dynamics models and simulators.

Figure 1

Pedestrian density map of the tracking area. The “corridor,” whose data were used for this work, is limited by the black lines. The map uses $A=0.25 {\mathrm{m}}^2$ sized cells, and averages over positions of pedestrians with velocity $v>0.5$ m/s, in order to compute density. The figure shows a time average over all data collected on a particular day, namely May 5th 2013.

Figure 2

Definition of the two- and three-person group observables.

Figure 3

$\rho$ dependence of pedestrian velocity. Individual pedestrians in black circles, two-person groups in red squares, three-person groups in blue triangles. Continuous lines provide standard error confidence intervals, while dashed lines provide a linear fit of the data (limited to the common definition range), given by a law $v^{\left(i\right)}={\alpha }_i+{\beta }_i\rho$, with ${\alpha }_1=1.286$ m/s, ${\beta }_1=-1.072 {\mathrm{m}}^3$/(ped s), determination coefficient $R_1^2=0.949$; ${\alpha }_2=1.147$ m/s, ${\beta }_2=-0.931 {\mathrm{m}}^3$/(ped s), $R_2^2=0.976$; ${\alpha }_3=1.092$ m/s, ${\beta }_3=-0.799 {\mathrm{m}}^3$/(ped s), $R_3^2=0.971$.

Figure 4

Black and circles: $\rho$ dependence of $(v^{\left(1\right)}-v^{\left(2\right)})/(v^{\left(2\right)}-v^{\left(3\right)})$, computed using the average values of Fig. 3. Dashed red: in order to decrease the effect of fluctuations, the same ratio is computed using the linear fits of Fig. 3.

Figure 5

$\rho$ dependence of $x_{\text{g2}}$ [Eq. (13)], in black circles, compared to a linear fit of the data (dashed red). Continuous lines provide standard error confidence intervals.

Figure 6

Probability distributions for $x_{\text{g2}}$ [Eq. (13)], in different $\rho$ ranges. The inset graph shows the position of the mode as a function of $\rho$. A one-way ANOVA analysis shows that the distributions in the figure are different in a statistically significant way ($p<{10}^{-8}$).

Figure 7

Probability distributions for $y_{\text{g2}}$ [Eq. (14)] in different $\rho$ ranges. The inset graph shows the position of the mode as a function of $\rho$. A one-way ANOVA analysis shows that the distributions in the figure are not different in a statistically significant way ($p\approx 0.35$).

Figure 8

Black circles: $\rho$ dependence of $x_{\text{g3}}$ [Eq. (15)], compared to a linear fit of the data (dashed red). Blue squares: $\rho$ dependence of $y_{\text{g3}}$ [Eq. (16)]. Continuous lines provide standard error confidence intervals.

Figure 9

Probability distributions for $x_{\text{g3}}$ [Eq. (15)] in different $\rho$ ranges. The inset graph shows the position of the mode as a function of $\rho$. A one-way ANOVA analysis shows that the distributions in the figure are different in a statistically significant way ($p<{10}^{-8}$).

Figure 10

Probability distributions for $y_{\text{g3}}$ [Eq. (16)] in different $\rho$ ranges. The inset graph shows the position of the mode as a function of $\rho$. A one-way ANOVA analysis shows that the distributions in the figure are not different in a statistically significant way ($p\approx 0.87$). On the other hand, the overall empirical distribution in $\rho <0.2$ ped/${\mathrm{m}}^2$ has an average value $140\pm 14$ mm, with a $p<{10}^{-8}$ value corresponding to the abreast configuration $y_{\text{g3}}=0$.

Figure 11

2D probability distributions for the position in a three-person group in a 2.5-m-wide square centered on $\mathbf{X}$. Left: $0\le \rho <0.05$; right: $0.1\le \rho <0.15$. The arrow shows the direction of motion of the group $\widehat{\mathbf{V}}$.

Figure 12

Black circles: $\rho$ dependence of $y_{g2}$ [Eq. (14)]. Continuous lines provide standard error confidence intervals. Red squares: $\rho$ dependence of $y_3-y_1$ for three-person groups. Dashed lines provide standard error confidence intervals.

Figure 13

Data points used for Figs. 3, 4, 5, 8, and 12. Black circles: number of individual pedestrians (i.e., pedestrians not explicitly coded as part of groups) for each density slot. Red squares: number of two-person groups. Blue triangles: number of three-person groups.

Figure 14

Red squares: number of two-person group data points (total number of observations) for each density slot. Blue triangles: number of three-person group data points (total number of observations) for each density slot.

Figure 15

Data points used for Figs. 6, 7, and 9–11. Red squares: number of two-person group data points (number of observations) for density slot. Blue triangles: number of three-person group data points (number of observations) for density slot.

Figure 16

Dependence of $x_{\text{g2}}$ on density. Black circles: $\rho$ computed using cells with $A=0.25 {\mathrm{m}}^2, T=300$ s, and data points with $v>0.5$ m/s (linear fit in dashed red). Blue squares: $\rho$ computed using cells with $A=24 {\mathrm{m}}^2, T=0.5$ s, and all data points (linear fit in dashed green, with $\alpha =0.614$ m, $\beta =-0.234 {\mathrm{m}}^3$/ped, and $R^2=0.831$).

Figure 17

Comparison between the empirical $p_E\left(r\right)$, theoretical $p_T\left(r\right)$, and calibrated $p_C\left(r\right)$ distributions in both data sets. For the ATC set, the black dotted line and circles represent $p_E\left(r\right)$, the black dashed line represents $p_T\left(r\right)$, and the black histogram represents $p_C\left(r\right)$. For the [14] set, the orange (gray) dotted line and squares represent $p_E\left(r\right)$, the orange dashed line represents $p_T\left(r\right)$, and the orange histogram represents $p_C\left(r\right)$. All data refer to densities in the $0\le \rho <0.05$ ped/${\mathrm{m}}^2$ range.

Figure 18

Comparison between the empirical $p_E\left(\theta \right)$, theoretical $p_T\left(\theta \right)$, and calibrated $p_C\left(\theta \right)$ distributions in both data sets. For the ATC set, the black dotted line and circles represent $p_E\left(\theta \right)$, the black dashed line represents $p_T\left(\theta \right)$, and the black histogram represents $p_C\left(\theta \right)$. For the [14] set, the orange (gray) dotted line and squares represent $p_E\left(\theta \right)$, the orange dashed line represents $p_T\left(\theta \right)$, and the orange histogram represents $p_C\left(\theta \right)$. All data refer to densities in the $0\le \rho <0.05$ ped/${\mathrm{m}}^2$ range, and they are limited to $0\le \theta <\pi$.