Scale-Free Correlation.

Figures5A and 5B show the relationship between the correlation function and distance. The number of individuals is 200. The blue dots show the correlation function for the orientation (using Equation (10)). The red dots show the correlation function of the speed (using Equation (12)). Both values gradually decrease as the distance becomes large. Especially the value of the speed correlation function suddenly rises up to positive values. In this case, the correlation lengths are  = 370 L (orientation) and  = 320 L (speed). Figures 5C and 5D show the relationship between the flock size and the correlation length. We ran simulations 100 times for flocks with 100, 200, and 300 individuals, respectively, and averaged the results over a certain interval (100 L on the horizontal axis). The error bars indicate the SD. The three-dimensional MTI model shows a scale-free correlation in both cases. The red dots correspond to the orientation in Figure 5C, and the blue dots correspond to the speed in Figure 5D. Both are well correlated (correlation coefficients of 0.87 for the orientation and 0.78 for speed). The gradients of both graphs are given as 0.36 (orientation) and 0.35 (speed).