  Collective motion of animal groups often undergoes changes due to
perturbations. 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.
