Fig. 5: Comparison of self-propelled particle model in presence or absence of MS.

We use a self-propelled particle model that particles follow the local interaction rules, i.e., AMS (a), ATHD (b) and average interaction (c). The additional potential well, imposed on one of individuals (red trajectories), aims to lead the flock come back to origin. The swarm size is 30 particles. The black points represent the end of trajectories. In (a), the flocking trajectories are generated by AMS with \(\alpha=0\). d–f The temporal curvature of flock trajectories respectively shown in (a–c). In (d–f), the solid curves represent the average of trajectory curvatures from 30 particles, and the shadow area represents the standard deviation (SD). g The distribution of nestedness of LF networks derived from flocking trajectories using AMS, ATHD and average interaction. The white points (or black lines) represent the median (or mean) value. h The distribution of Spearman correlation (\(\rho\)) between LF and MS over different combinations of \(t\) and \(\tau\) from flocking trajectories using AMS, ATHD and average interaction. For each flock we take 10 time stamps for \(t\) and 11 time points for \(\tau\) to perform the correlation analysis between LF and MS. In (g) and (h), we run 50 independent simulations for each interaction type.