Scale-free correlations in starling flocks

Abstract

From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behavior have aimed to understand how a globally ordered state may emerge from simple behavioral rules. Less effort has been devoted to understanding the origin of collective response, namely the way the group as a whole reacts to its environment. Yet, in the presence of strong predatory pressure on the group, collective response may yield a significant adaptive advantage. Here we suggest that collective response in animal groups may be achieved through scale-free behavioral correlations. By reconstructing the 3D position and velocity of individual birds in large flocks of starlings, we measured to what extent the velocity fluctuations of different birds are correlated to each other. We found that the range of such spatial correlation does not have a constant value, but it scales with the linear size of the flock. This result indicates that behavioral correlations are scale free: The change in the behavioral state of one animal affects and is affected by that of all other animals in the group, no matter how large the group is. Scale-free correlations provide each animal with an effective perception range much larger than the direct interindividual interaction range, thus enhancing global response to perturbations. Our results suggest that flocks behave as critical systems, poised to respond maximally to environmental perturbations.

Fig. 1.

(A) 2D projection of the velocities of the individual birds within a starling flock at a fixed instant of time (flock 28-10; 1,246 birds, linear size L = 36.5 m). Vectors are scaled for clarity (see Dataset S1 for original data). The flock is strongly ordered and the velocities are all aligned. (B) 2D projection of the individual velocity fluctuations in the same flock at the same instant of time (vectors scaled for clarity). The velocity fluctuation is equal to the individual velocity minus the center of mass velocity, and therefore the spatial average of the fluctuations must be zero. Two large domains of strongly correlated birds are clearly visible. (C) Normalized probability distribution of the absolute value of the individual velocities and of the absolute value of the velocity fluctuations (same flock as in A and B). The velocity fluctuations are much smaller in modulus than the full velocities.

Fig. 2.

(A) The correlation function C(r) is the average inner product of the velocity fluctuations of pairs of birds at mutual distance r. This correlation function therefore measures to what extent the orientations of the velocity fluctuations are correlated. The function changes sign at r = ξ, which gives a good estimate of the average size of the correlated domains (flock 28-10). (B) The correlation function C_sp(r), on the other hand, measures the correlations of the fluctuations of the modulus of the velocity, i.e., the speed. This correlation function measures to what extent the variations with respect to the mean of the birds’ speed are correlated to each other. The speed correlation function changes sign at a point r = ξ_sp, which gives the size of the speed-correlated domains (flock 28-10). Both correlation functions in A and B are normalized to give C(r = 0) = 1. (C) The orientation correlation length ξ is plotted as a function of the linear size L of the flocks. Each point corresponds to a specific flocking event and it is an average over several instants of time in that event. Error bars are SDs. The correlation length grows linearly with the size of the flock, ξ = aL, with a = 0.35 (Pearson's correlation test: n = 24, r = 0.98, P < 10⁻¹⁶), signaling the presence of scale-free correlations. (D) Also in the case of the correlation function of the speed, the correlation length ξ_sp grows linearly with the size of the flock, ξ_sp = aL, with a = 0.36 (Pearson's correlation test: n = 24, r = 0.97, P < 10⁻¹⁵). Error bars are SDs.

Fig. 3.

(A) The correlation functions of several flocks are plotted vs. the rescaled variable x = r/ξ. (Inset) The modulus of the derivative of the correlation function with respect to the rescaled variable x, evaluated at x = 1, plotted vs. the correlation length ξ for all flocking events. The derivative is almost constant with ξ, indicating that the exponent γ in the scale-free asymptotic correlation is very close to zero. The black and red lines represent the best fits to, respectively, a constant and a logarithm (see text). (B) Same as in A for the speed correlation.

Fig. 4.

Random synthetic velocities. In each flock we replace the actual birds’ velocity fluctuations with a set of synthetic random vectors correlated over a length λ that we can arbitrarily tune (see text). The synthetic fluctuations are located at the same positions as birds in a real flock (we used flock 28-10, the same as in Fig.1). (A) Synthetic fluctuations in the non-scale-free case, λ = 0.05L. The domains are quite small and have a size comparable to λ. (B) Synthetic fluctuations for λ = 4L. In this scale-free limit the domains are very similar to the actual biological ones displayed in Fig.1B. (C) Synthetic correlation functions for various values of the decay length λ. By increasing λ the synthetic correlation function becomes more and more long ranged and it finally becomes very close to the actual biological one in the scale-free regime λ > L. (D) Synthetic correlation length ξ_SYNTH, as a function of the decay length λ (each point is an average over 50 synthetic samples; errors bars are SDs). As long as λ is smaller than the size of the flock L, ξ_SYNTH grows following λ. However, in the scale-free regime, λ > L, ξ_SYNTH saturates to a value very close to the actual biological one.