Avalanches of choice: how stranger-to-stranger interactions shape crowd dynamics

TL;DR

The study investigates how interactions among strangers shape pedestrian routing at a bifurcation in a real-world transit setting, using a three-year, high-resolution trajectory dataset from Eindhoven Centraal. It reveals a strong stranger-following effect, where individuals imitate the path chosen by the person immediately ahead, producing bursty, avalanche-like cascades in path use. A cost-based routing model incorporating a stranger-following reward term, speed variability, and herding captures the observed patterns, with the imitation term dominating other factors. These findings imply that local, anonymous social imitation can drive suboptimal crowd flows and offer actionable insights for crowd management and urban design, supported by methods to infer group composition from path-choice statistics and by robust, long-term observational data.

Abstract

Pedestrian routing choices play a crucial role in shaping collective crowd dynamics, yet the influence of interactions among unfamiliar individuals remains poorly understood. In this study, we analyze real-world pedestrian behavior at a route split within a busy train station using high-resolution trajectory data collected over a three-year time frame. We disclose a striking tendency for individuals to follow the same path as the person directly in front of them, even in the absence of social ties and even when such a choice leads to a longer travel time. This tendency leads to bursty dynamics, where sequences of pedestrians make identical decisions in succession, leading to strong patterns in collective movement. We employ a stochastic model that includes route costs, randomness, and social imitation to accurately reproduce the observed behavior, highlighting that local imitation behavior is the dominant driver of collective routing choices. These findings highlight how brief, low-level interactions between strangers can scale up to influence large-scale pedestrian movement, with strong implications for crowd management, urban design, and the broader understanding of social behavior in public spaces.

Figures (13)

• Figure 1: Real-life measurement setup and pedestrian flow patterns at Eindhoven Centraal Railway Station (The Netherlands). (A) Experimental layout of the platform. Passenger trajectories are recorded using a commercial tracking system. Disembarking passengers from Track 4 exit the train via three door locations---$L_1$, $L_2$, and $L_3$---and encounter a path bifurcation, choosing between Path A and Path B. A large indoor waiting area (Kiosk) is present on the platform, along with other marked elements such as elevators, staircases, and benches. Heat maps of pedestrian positions after exiting through doors (B) $L_1$, (C) $L_2$, and (D) $L_3$, respectively. Colors represent the density of observed positions on a logarithmic scale based on a 2D count histogram. Overlaid streamlines show the mean velocity vector field (spatially binned), highlighting the most probable pedestrian trajectories.
• Figure 1: Definition of the order of the exiting passenger. The black dots denote the people who have already standing on the platform. The red dots denote the passengers who just get off the train. The blue circle represents the detection boundary based on which the order of exiting passenger is defined.
• Figure 2: Statistical analysis of pedestrian path-choice behavior. (A-C) Path-choice probabilities for passengers exiting through doors (A) $L_1$, (B) $L_2$, and (C) $L_3$, plotted against the number of pedestrians on the shorter path (Path A). We report the marginal probability $P(X_i=\chi)$, the conditional probability $P(X_i=\chi|X_{i-1}=\chi)$, and the probability excluding groups $P(X_i=\chi|X_{i-1}=\chi, \neg G_i^{(2)})$, with $\chi=A$ (green colors) and $\chi=B$ (blue color). The observation of $P(X_i=B|X_{i-1}=B,\neg G_i^{(2)})) \approx P(X_i=B|X_{i-1}=B) > P(X_i=B)$ shows a clear evidence of stranger-following effect. (D) Group ratio $P(G)$ as a function of the distance threshold with covariance of speeds $\mathrm{Cov}(v_i,v_j)\!>\!0.3$ and mean directional alignment $\langle \cos\theta\rangle_t\!>\!0.9$, for door position of $L_1$ (blue), $L_2$ (orange), and $L_3$ (green), respectively. In this work, the group detection are based on the thresholds $\left\langle d_{i,j} \right\rangle_t = 1.5~[m]$, $\mathrm{Cov}(v_i,v_j)=0.3$, and $\left < \cos \theta \right >_t = 0.9$. The corresponding example trajectories of pedestrians identified as belonging to the same group (same color; solid vs. dashed for different members) are shown in panel (E), with triangles and circles marking start and end points, respectively. For simplicity, we present only groups of size two, which make up the majority of all observed groups. (F-H) Robustness of the stranger-following effect: $P(X_i{=}B \mid X_{i-1}{=}B,\, \neg G_i^{(2)})$ computed under two sets of thresholds for the group criteria: (light blue) a reasonable set ($\langle d_{i,j}\rangle_t{=}1.5$ m, $\mathrm{Cov}(v_i,v_j){=}0.3$, $\langle \cos\theta\rangle_t{=}0.9$) and (red) a deliberately lenient set ($\langle d_{i,j}\rangle_t{=}3.0$ m, $\mathrm{Cov}(v_i,v_j){=}0.1$, $\langle \cos\theta\rangle_t{=}0.5$). The relation $P(X_i{=}B \mid X_{i-1}{=}B,\, \neg G_i^{(2)}) \approx P(X_i{=}B \mid X_{i-1}{=}B) > P(X_i{=}B)$ persists, indicating robustness to reasonable variations in detection thresholds, provided the threshold is within a reasonable range and applied uniformly across door locations.(I) Comparison of group ratios obtained in three ways for each door: (i) conditioned estimates from path-choice statistics $P(G_i^{(2)}\mid X_{i-1}{=}A)$ (blue) and $P(G_i^{(2)}\mid X_{i-1}{=}B)$ (orange), (ii) the resulting weighted estimate $P(G)$ (green), and (iii) direct detection based on the three criteria (red). Overall path-choice probabilities $P(X_i{=}A)= 0.967, 0.855, 0.707$ for $L_1$, $L_2$, and $L_3$, respectively.
• Figure 2: Group detection criteria. The detected group ratio as a function of the chosen threshold based on the criteria of (a) average coexistence distance; (b) velocity covariance; and (c) directional alignment for door position of $L_1$ (blue), $L_2$ (orange), and $L_3$ (green), respectively.
• Figure 3: Theoretical routing model results of path choice. (A) Fundamental diagram showing the relationship between the average walking speed and the number of pedestrians observed within the same fixed-size segment (purple-shaded regions in Fig. \ref{['FIG1']}A) on Path A ($N_A$, orange circles) and Path B ($N_B$, blue circles). The two regions have identical length and width; therefore, the counts directly represent comparable local crowding conditions. The dashed black line represents a linear fit of the form $V_i = V_0 - \kappa N_i, \quad i \in \{A, B\},$, with $V_0 = -0.007$ m/s and $\kappa = 1.2785$ m/s. Error bars denote standard deviations. (B-G) Model-data comparison of the conditional probability excluding groups of choosing Path A (panels B, D, F) and Path B (panels C, E, G) as a function of the number of pedestrians on Path A, for exit doors $L_1$, $L_2$, and $L_3$. Black symbols: experiments. Grey dashed: time-minimization-only baseline ($t_i$). Grey dotted: time-minimization-only $+$ speed variability ($S_i \cdot t_i$). Orange dash-dotted: herding only ($f_i \cdot t_i$). Yellow dashed: herding $+$ speed variability ($r_i^{(\chi)} \cdot S_i \cdot t_i$). Blue dotted: following stranger only ($r_i^{(\chi)} \cdot t_i$). The following-stranger-only model (blue) already captures the main experimental trends, while adding herding and speed variability provide only minor local adjustments, showing that the stranger-following effect is an essential component of pedestrian decision-making. Only the following-stranger-only model (blue) is shown here for clarity; additional models that include stochasticity and/or penalty terms are provided in SI Appendix. For all the modeling results shown in (B-G), (i) the penalty function in Eq. \ref{['eq:penalty']} is with a same set of parameters (we apply the penalty function only to $T_i$ on Path A), i.e., $a=0.1, s=20, n_0=0.2, f_A=1$; (ii) the reward function in Eq. \ref{['eq:reward']} is with the parameter $\Gamma = 0.9$.