Universal structural mechanism underlying collective order in active matter

How collective order in active matter arises from simple local interactions remains a central and fascinating question. While previous theories have primarily focused on how control parameters drive ordering, the underlying structural mechanisms remain elusive. Here, we uncover a universal structure–order relationship in Vicsek systems: across a range of densities and noise strengths, the polar order parameter exhibits a general correlation with the average interaction number. This relationship reconciles the apparently distinct order transitions induced by different control parameters within a unified framework of structure–order evolution, comprising four stages: disorder, order growth, order coherence, and order saturation. We further identify scale-invariant cluster structures and demonstrate that both cluster-level and system-level degree distributions converge toward a universal form, and similar cluster structures are identified in starling flocks. Building on these structural universalities, we develop a theoretical framework that predicts the order parameter from structural statistics, highlighting the vital role of structural evolution in the onset, growth, and saturation of the order parameter. This work provides a new structural perspective on the disorder–order transition of Vicsek systems and suggests new principles for understanding and engineering collective behavior in active matter.