Minority-Triggered Reorientations Yield Macroscopic Cascades and Enhanced Responsiveness in Swarms

TL;DR

This work introduces a simple, biologically plausible mechanism: a minority-triggered reorientation rule that qualitatively changes the macroscopic system behavior compared to traditional flocking models, as it generates heavy-tailed cascades of reorientations over broad parameter ranges.

Abstract

Collective motion in animals and cells often exhibits rapid reorientations and scale-free velocity correlations. This allows information to spread rapidly through the group, allowing an adequate collective response to environmental changes and threats such as predators. To explain this phenomenon, we introduce a simple, biologically plausible mechanism: a minority-triggered reorientation rule. When local order is high, agents sometimes follow a strongly deviating neighbor instead of the majority. This rule qualitatively changes the macroscopic system behavior compared to traditional flocking models, as it generates heavy-tailed cascades of reorientations over broad parameter ranges. Our mechanism preserves cohesion while markedly enhancing collective responsiveness because localized directional cues elicit amplified group-level reorientation. Our results provide a parsimonious, biologically interpretable route to critical-like fluctuations and high responsiveness during flocking.

Figures (11)

• Figure 1: Minority interaction leads to avalanches. (a) In standard Vicsek dynamics, a particle $i$ (black) aligns with the average orientation of its neighbors (green). With the minority interaction, if the surrounding neighborhood is sufficiently ordered (condition 1), the same particle may instead follow a defector (red) that strongly deviates from the local consensus (condition 2). The outlined red arrow indicates the focal particle's orientation after reorienting due to the minority interaction. (b) The minority interaction creates perturbations that propagate through the system as an avalanche of reorientations, temporarily disrupting global order. Time evolution of the order parameter $\phi(t)$ in our model with minority interaction ($L=32, \rho=1, \gamma=-0.3, \epsilon=0.6, \eta=0.1$), showing large fluctuations corresponding to avalanche events. The dashed line ($\phi_c$) indicates the threshold used to define avalanche duration $\tau$ and size $\Delta \phi$. The duration of an avalanche $\tau$ is defined as the time period during which the order parameter is below this threshold and the avalanche size is the maximum deviation of the order parameter from $\phi_c$, see inset. In contrast, the order parameter for the classical Vicsek model with identical parameters (gray line) shows much smaller fluctuations. (c) The direction of the average velocity, $\cos(\Theta(t))$, also exhibits large fluctuations coinciding with the avalanches in $\phi(t)$ for the minority model (red), whereas it remains relatively stable for the standard VM (gray). (d)-(f) Snapshots of an example simulation, where a single defector (red circle in d) causes an avalanche of reorientations (red circles in e and f) in an initially ordered flock.
• Figure 2: Avalanche statistics. Complementary cumulative distribution functions (CCDF), $P(X \ge x)$. (a) Avalanche size $\Delta\phi$ for different minority interaction thresholds ($\epsilon, \gamma$) compared to the standard VM (gray). System parameters: $L=32, \rho=1.0, \eta=0.1$. (b) Avalanche duration (return time $\tau$) for the same parameters as (a). (c) Avalanche size CCDF for fixed thresholds ($\epsilon=0.3, \gamma=-0.6$) and varying system sizes $L$. (d) Avalanche duration CCDF for the same parameters as (c). Avalanches can be macroscopic and orders of magnitude larger than in the standard VM.
• Figure 3: Spatial velocity correlations. (a) Velocity fluctuation correlation function $C(d)$ versus inter-particle distance $d$ for systems with different sizes $L$ (indicated by color). Solid lines: model with minority interaction. Dashed lines: corresponding standard VM. Correlations are significantly stronger with the minority interaction. (b) Correlation length $d_0$ (defined as $C(d_0)=0$) scales linearly with system size $L$ for the minority model (black circles) and the VM (gray circles). Dashed lines are linear fits. Model parameters identical to those in Fig. \ref{['fig:avalanche_stats']}c,d.
• Figure 4: Parameter dependence of avalanche behavior. Heatmap showing (a) the time-averaged order parameter $\ev{\phi}$ and (b) the variance ratio $\text{Var}(\phi) / \text{Var}(\phi_{\text{VM}})$ across different values of minority interaction thresholds $\epsilon$ and $\gamma$. System parameters: $L=32, \rho=1.0, \eta=0.1$. A variance ratio of one corresponds to behavior identical to the standard VM, while higher values indicate stronger avalanche activity. We observe 10 to 1000 times higher variance over a large parameter range, indicating that the avalanche behavior does not require fine-tuning. For large $\gamma$ and small $\epsilon$ (lower right corner), avalanches become so frequent that they notably reduce the average order (a). The blue, green, and red dots correspond to the parameter combinations used for the lines of the same color in Fig. \ref{['fig:avalanche_stats']}a,b.
• Figure S1: Collective responsiveness to a controlled directional perturbation. Time evolution of the order parameter $\phi(t)$ (blue) and mean orientation $\cos(\Theta(t))$ (orange) after a single-particle perturbation: one particle is forced to orient at $\theta = \pi$ (opposite to the direction of a perfectly aligned flock; see Fig. \ref{['fig:minority_interaction']}d-f for a sketch of the setting) for $\tau = 5$ time steps (dashed vertical line), then released. (a) Standard Vicsek model. (b)--(i) Model with minority interaction for eight combinations of $(\epsilon, \gamma)$. The VM rapidly re-aligns with negligible directional change. Across a broad range of minority interaction parameters, the localized perturbation is amplified into a group-wide reorientation event. System parameters: $L=32$, $N=200$, $v_0=0.5$, $\eta=0.1$; single representative realizations.