Quantitative relationship between coupled structural dynamics and collective order in the original Vicsek model

Abstract While the Vicsek model is widely used to understand active systems, the explicit relationship between structural evolution and the order transition remains an open question. Here we investigate this relationship in the original Vicsek system. We uncover a structure–order relationship across the investigated densities and noise strengths at fixed velocity and particle number, in which the polar order parameter exhibits a general correlation with the average interaction number. Furthermore, we reveal underlying structural self–similarities: cluster–level and system–level interaction number distributions show consistent normalized profiles, alongside a correlation between cluster size and orientation. Building on these correlations, we develop a phenomenological framework that quantitatively relates the order parameter to structural statistics. Moreover, over the investigated range of velocities, the structural features are maintained, whereas the structural–order correlation is augmented by a velocity factor. This work provides a structural perspective on the Vicsek model and suggests a quantitative relationship between microscopic structural organization and macroscopic collective behavior.