Probabilistic dynamics of small groups in crowd flows

TL;DR

The paper tackles the probabilistic dynamics of dyads in real crowds by leveraging an unprecedented dataset of over $6\,M$ dyads drawn from roughly $21\,M$ trajectories at Eindhoven Central Station. It introduces Orientation Log-Odds $\Pi_{\text{OLO}}=\log_2\frac{\mathbb{P}(\Gamma^{\leftrightarrow})}{\mathbb{P}(\Gamma^{\uparrow\downarrow})}$ to quantify the likelihood of abreast versus in-file formation and demonstrates that its derivative with respect to dyad speed can be expressed as the product of two velocity-density fundamental diagrams. Across regimes—from free flow to standing crowds and generic co- and counter-flow—the authors show that dyad velocity and formation are governed by density and relative crowd motion, with systematic crossovers: abreast dominates at low density but in-file becomes more probable as density rises, especially under opposing or standing crowds. The resulting data-driven, probabilistic description enables integration of dyad-aware dynamics into microscopic crowd simulations and macroscopic active-matter models, contributing toward more accurate and robust crowd dynamics modeling with explicit group structure.

Abstract

Pedestrians in crowds frequently move as part of small groups, constituting up to 70% of individuals. Dyads (groups of two) are the most frequent. Understanding quantitatively the dynamics of dyads walking in crowds is therefore an essential building block towards a fundamental comprehension of crowd behavior as a whole, and mandatory for accurate crowd dynamics models. Unavoidably, due to the non-deterministic behavior of pedestrians, characterizations of the dynamics must be probabilistic. In this work, we analyze the dynamics of over 6M dyads: a statistical ensemble of unprecedented resolution within a multi-year real-life pedestrian trajectory measurement campaign (21M trajectories, from Eindhoven Station, NL). We provide phenomenological models for dyad behavior in dependence of the surrounding crowds state. We present a thorough collection of fundamental diagrams that probabilistically relate both dyad velocity and formation to the state of the surrounding crowd (density, relative velocity). Depending on the surrounding crowd, dyads adjust interpersonal distance and may shift in formation, possibly turning from abreast states (which favors social interaction) to in-line (which favors navigationing dense crowds). To quantitatively investigate formation changes, we introduce a scalar indicator, which we dub Orientation Log-Odds (OLO), that quantifies the relative log-likelihood of abreast versus in-file formations. Conceptually, the OLO quantifies energy difference of the abreast vs. in-file configuration under a Boltzmann-like assumption. We model how OLO depends on the crowd state, showcasing that its derivative is a product of two velocity-density fundamental diagrams. Together, these results provide a statistically robust, data-driven description of dyad configuration dynamics in real-world crowds, establishing a foundation towards new predictive, group-aware crowd models.

Figures (13)

• Figure 1: Example dyad trajectory and definition of dyad observables. (a) Section of Eindhoven Central Station platform 3-4 with an example trajectory (red and orange lines) of a dyad entering from the bottom left and moving toward the top right of this domain. The red and orange markers show the dyad member positions at $5~\mathrm{s}$ intervals. Other pedestrians positions (blue) are from a snapshot at time $20~\mathrm{s}$; at this time the dashed circle marks a $2~\mathrm{m}$ proximity radius around the dyad center of mass $\boldsymbol{r}_{\text{com}}$ with pedestrians inside colored in green. (b) Observables in the dyad's frame of reference. The dyads frame of reference is defined by the basis $(\textbf{e}_{\parallel}, \textbf{e}_{\perp})$ (Eq \ref{['eq:dyadframe']}), where the centre-of-mass velocity $\textbf{v}_{com}$ (Eq \ref{['eq:basicdyadstuff']}) is parallel to $v_{||}$. Besides $\textbf{v}_{com}$, the interpersonal distance $d_{ij}$, and the dyad angle $\phi=-\operatorname{atan2}(x_r,y_r)$ are illustrated, where $\operatorname{atan2}$ is the two-argument arctangent function. Considered observables that depend on crowd members within the radius $R$ are their mean velocity $\textbf{v}_{prox}$ and local density $\rho$ (Eq \ref{['eq:density']}, here $\rho = 0.56\,\mathrm{p/m}^2$. (c) Dyad-crowd interaction regimes $B$ (Eq \ref{['eq:binningdef']}) defined in dependence on $\textbf{v}_{prox}$ and the angle $\alpha$. We shall consider dynamics on only in free-flow (no crowd), co-, counter-flow and with standing crowd. (d) $\phi$ and $\rho$ over time as the dyad in (a) traverses the domain, highlighting the abreast (red) and in-file (green) regions. The vertical lines correspond time snapshots in (a). Between $t=15\,\mathrm{s}$ and $t=35\,\mathrm{s}$ the dyad enters a region of higher $\rho$, temporally going in in-file formation before going back to an abreast formation.
• Figure 2: Dependency of dyad speed, $\text{v}_{\text{com}}$, on crowd density, $\rho$, for abreast (triangles) and in-file (circles) configurations. (a) Fundamental diagram ($\mathbb{E}[\text{v}_{\text{com}} | \rho,\,\textbf{r}_{loc }]$): mean velocity and standard deviation vs. local density for non-dyad crowd members (blue squares), abreast dyads $\text{v}_{\text{com}}^{\leftrightarrow}$ (Eq \ref{['eq:speed_abreast']}, green triangles), and in-file dyads $\text{v}_{\text{com}}^{\updownarrow}$ (Eq \ref{['eq:speed_infile']}, red circles). (b) Walking speed probability density in free-flow conditions ($\mathbb{E}[\text{v}_{\text{com}} | B_{\text{free}}]$) for dyad members (yellow circles) and non-dyad pedestrians (blue squares). Dashed vertical line: modal speed; dotted line with cross: mean speed; solid line: Gaussian fit to modal region. Both modal and mean speeds are slower for dyads vs. non-dyads. (c) Abreast dyad speeds conditioned on $B$ Eq \ref{['eq:binningdef']} ($\mathbb{E}[\text{v}_{\text{com}} | B,\,\rho]$). Note the different y-axis scale from (a). Co-moving dyads ($B_{\text{coflow}}$, dashed blue) maintain highest speeds, counter-flow dyads ($B_{\text{counterflow}}$, dashed yellow) intermediate speeds, and dyads in stationary crowds ($B_{\text{standing}}$, dashed gray) lowest speeds. (d) In-file dyad speeds (solid lines) across same $B$ as in (c). The ordering of speeds among the different dyad types is the same as in (c). At low densities, in-file dyads are slower than abreast; this reverses at higher densities. Crossover occurs at $\rho \approx 0.5\,\mathrm{p/m}^2$ for $B_{\text{standing}}$ and $B_{\text{counterflow}}$, and at $\rho \approx 0.8\,\mathrm{p/m}^2$ for $B_{\text{coflow}}$.
• Figure 3: Probability density of dyad positions arranged in fundamental diagram format, showing how dyad configurations vary with density and speed. The colorization is scaled by the minimum and maximum values of the pdf indicated by Count[min, max] in each subplot. At low densities (left columns), abreast configurations near the $x_r$ axis dominate across all speed bins (rows). As density increases, a secondary peak emerges near the $y_r$ axis, corresponding to in-file configurations. This shift indicates that the likelihood of dyad configurations depends systematically on crowd density.
• Figure 4: Probability density of dyad positions under different dyad-crowd interaction regimes $B$ (Eq \ref{['eq:binningdef']}), showing how dyad configurations depend on relative crowd motion and density. Minimum and maximum values are indicated by Count[min, max] in each subplot. Top row: $B_{\text{coflow}}$ (co-moving), middle row: $B_{\text{standing}}$ (stationary crowd), bottom row: $B_{\text{counterflow}}$ (counter-flow). In contrast to Fig \ref{['fig:complexfund']}, this analysis shows that in-file configurations dominate at higher densities (right column) when dyads traverse standing crowds or move against the flow. At comparable densities, dyads moving with the crowd remain predominantly abreast, while those in standing or opposing crowds adopt in-file arrangements.
• Figure 5: Dependence of density $\rho$ and dyad speed $\text{v}_{\text{com}}$ on the relative position $\textbf{r}_{loc }$. (a) Density-configuration ($\mathbb{E}[\rho \mid (x_r, y_r)]$) diagram indicating that abreast configurations occur predominantly at lower $\rho$ than in-file configurations. (b) Speed-configuration ($\mathbb{E}[\text{v}_{\text{com}} \mid (x_r, y_r)]$) diagram showing that smaller interpersonal distances correspond to lower speeds in both configurations, with a stronger effect for in-file dyads. The highest expected $\text{v}_{\text{com}}$ is observed in a concentrated region centered along the $y_r$ axis.