A Theoretical Framework for the Formation of Large Animal Groups: Topological Coordination, Subgroup Merging, and Velocity Inheritance

Abstract

Large animal groups -- bird flocks, fish schools, insect swarms -- are often assumed to form by gradual aggregation of sparsely distributed individuals. Using a mathematically precise framework based on time-varying directed interaction networks, we show that this widely held view is incomplete. The theory demonstrates that large moving groups do not arise by slow accumulation; instead, they emerge through the rapid merging of multiple pre-existing subgroups that are simultaneously activated under high-density conditions. The key mechanism is topological: the long-term interaction structure of any moving group contains a single dominant strongly connected component (SCC). This dominant SCC determines the collective velocity -- both speed and direction -- of the entire group. When two subgroups encounter one another, the trailing subgroup aligns with -- and ultimately inherits -- the velocity of the dominant SCC of the leading subgroup. Repeated merging events naturally generate large groups whose speed is predicted to be lower than the mean speed of the original subgroups. The same dynamics explain several universal empirical features: broad neighbour-distance distributions, directional asymmetry in neighbour selection, and the characteristic narrow-front, wide-rear geometry of real flocks. The framework yields testable predictions for STARFLAG-style 3D datasets, offering a unified explanation for the formation, maintenance, and geometry of coordinated animal groups.