Marginal speed confinement resolves the conflict between correlation and control in collective behaviour

Abstract

Speed fluctuations of individual birds in natural flocks are moderate, due to the aerodynamic and biomechanical constraints of flight. Yet the spatial correlations of such fluctuations are scale-free, namely they have a range as wide as the entire group, a property linked to the capacity of the system to collectively respond to external perturbations. Scale-free correlations and moderate fluctuations set conflicting constraints on the mechanism controlling the speed of each agent, as the factors boosting correlation amplify fluctuations, and vice versa. Here, using a statistical field theory approach, we suggest that a marginal speed confinement that ignores small deviations from the natural reference value while ferociously suppressing larger speed fluctuations, is able to reconcile scale-free correlations with biologically acceptable group's speed. We validate our theoretical predictions by comparing them with field experimental data on starling flocks with group sizes spanning an unprecedented interval of over two orders of magnitude.

Fig. 1. Qualitative sketch of linear vs. marginal speed-restoring force.

In the linear case the force pulls the speed back to its natural reference value v₀ proportionally to the deviation from v₀. Instead, in the marginal case, the force is extremely weak for small deviations from the reference speed, while it increases very sharply for large deviations, harshly suppressing them.

Fig. 2. Experimental evidence on starling flocks.

a The equal-time space correlation function of the speed fluctuations (for the definition of the correlation function see the “Methods” section), plotted against the distance r between the birds rescaled by the flock’s size L, for some typical flocks (each colour corresponds to a different flock); the fact that all the curves collapse onto each other indicates that the spatial range of the speed correlation, namely the correlation length ξ_sp, scales with L, i.e. that the system is scale-free (see also Fig. 3a, c). b Scatter plot displaying polarization vs. mean speed of each flock in all recorded events; as the polarization, $\mathrm{\Phi }=\left(1/N\right)∣{\sum }_i{\mathbf{v}}_i/v_i∣$, is quite close to 1 for all flocks, it is more convenient to plot 1−Φ in log scale. Data show that starling flocks are highly ordered systems, incompatible with the standard notion of near-criticality (green points correspond to medians over time, error bars to median absolute deviations). The probability distributions of polarization and mean speed are reported in panels c and d, showing that the typical mean group’s speed is 12 m s⁻¹, with fluctuations of about 2 m s⁻¹.

Fig. 3. Linear vs. Marginal speed control.

a Natural flocks show a clear scale-free behaviour of the speed correlation length, ξ_sp, which scales linearly with L (Pearson coefficient r_P = 0.97, p < 10⁻⁹). SPP simulations with linear speed control yields scale-free correlations over the entire range of L only at the smallest value of the stiffness g (dark red). b Natural flocks show no detectable dependence of their mean speed on the number of birds in the flock (Spearman coefficient r_S = −0.13, p = 0.21; the black line is the average over all flocks). SPP simulations with linear control give a near-constant speed compatible with experiments only at the largest value of the stiffness g (light yellow); coloured lines represent the theoretical prediction of Eq. (6). Linear speed control is therefore unable to reproduce both experimental traits at the same time. c The correlation length in SPP simulations with marginal speed control scales linearly with L over the full range, provided that the temperature/noise T is low enough to have a polarization equal to the experimental one. d At the same value of the parameters as in panel c, SPP simulations with marginal control give mean group’s speed very weakly dependent on N, fully compatible with the experimental data; the y scale of the main plot is set to make a comparison with panel b, while in the inset we report the same data over a smaller y range to appreciate the agreement between theory (blue line) and simulations, and to show that the trend of group’s speed with N of the marginal model is really weak. [Each length scale ℓ pertaining to numerical simulations is reported in meters by normalizing it to the inter-particle distances, $\ell ={\ell }^{~}\left(r_1^{\exp }/r_1^{~}\right)$, where $r_1^{\exp }$ and $r_1^{~}$ are the mean inter-particle distances of experiments and simulations respectively. Numerical and experimental correlation lengths are reported on the same scale by matching the curves at the scale-free value of the parameters; numerical and experimental speeds are reported on the same scale by matching the curves at the largest value of N. Coloured points correspond to averages over numerical data, error bars to standard deviations. Black points correspond to the median (over time) of experimental data for each individual flocking event, error bars to median absolute deviations].

Fig. 4. Single particle speed distributions.

We measured the single particle speed distribution (as opposed to the mean speed of the group of Fig. 3), for different groups sizes, for real flocks (grey), the linear model (dark red) and the marginal model (light blue). The parameters in the marginal case are the same as in Fig. 3, while for the linear case we fixed the stiffness at g = 10⁻³, which is the only value giving scale-free correlations at all sizes (see Fig. 3). The individual speed on the abscissa has been normalized to the reference speed value v₀, in order to compare all cases on the same plot. The data show that linear speed control is unable to fit the experimental distribution even at the largest N, and it is a total disaster in the medium and smallest N cases. On the other hand, the marginal theory fits the data in a rather satisfactory way at all values of N.