Identifying manifolds underlying group motion in Vicsek agents

In a topological sense, we describe these changes as switching between low-dimensional embedding manifolds underlying a group of evolving agents. To characterize such manifolds, first we introduce a simple mapping of agents between time-steps. Then, we construct a novel metric which is susceptible to variations in the collective motion, thus revealing distinct underlying manifolds. The method is validated through three sample scenarios simulated using a Vicsek model, namely switching of speed, coordination, and structure of a group. Combined with a dimensionality reduction technique that is used to infer the dimensionality of the embedding manifold, this approach provides an effective model-free framework for the analysis of collective behavior across animal species. In animal groups, the response to a perturbation--internal or external--is often manifested in the form of changes in group speed, coordination, or structure [3,5,11,16,27]. Such changes are witnessed in fish schools and bird flocks under attack [15,17,22], foraging animal groups [4, 8], and human crowds exposed to alarm situations leading to panic [12, 19]. Based on our recent effort demonstrating that collective motion is associated with a low-dimensional embedding [1, 2, 6, 7, 10], we expect that such behavioral changes should be manifested in variation of the topology of an underlying manifold.