Fig. 1: The leader-follower relations of different patterns of collective motions.

The flocking trajectories are from three datasets: mobbing (a), circling (b), and transit (c), which are categorized into maneuver (mobbing and circling) and smooth (transit) modes of collective motions. In (a–c), the gradient color from blue to red corresponds to the recording time from beginning to end. d–f The LF relation matrix and corresponding LF network of three flocks shown in panels a–c, respectively. The negative values (\({\tau }_{{ij}}^{{{{{{\rm{LF}}}}}}} < 0\)) in LF relation matrix represent that individual-\(j\) leads individual-\(i\), and equals the directed edges pointing from leader-\(j\) to follower-\(i\) in the corresponding LF networks. The box colors from black to white correspond to the descending order of absolute value of \({\tau }_{{ij}}^{{{{{{\rm{LF}}}}}}}\). The axis in (d–f) is ordered by the nestedness of their corresponding LF networks. The nestedness of LF networks as a function of average order for three flocking datasets of mobbing (g), circling (h), and transit (i). The right marginal plots show the nestedness distribution of LF networks. The order parameter is a temporal measurement evaluating the motion polarization of a flock at time \(t\), i.e., \({\sum }_{i=1}^{N}{\hat{{{{{{\bf{v}}}}}}}}_{i}(t)/N\), and the average order is the mean of polarization over the whole time. In (d–i), the LF relation matrices are calculated from the whole period of each flock, and the nestedness is measured by NODF. In (g–i), each point represents a flock from three datasets, and the nonparametric regression and bootstrap sampling are performed to calculate the trend (red curve) and its 94% confidence interval (red shadow) between nestedness and average order.