Persistent homology as a universal metric for collective motion

From flocking birds to human crowds, collective motion patterns arise from the coordinated motion of individuals in a group. These patterns are notoriously hard to infer from the actions of each individual, yet are clearly distinguishable even by an untrained observer. Established formal approaches towards analysing collective motion patterns rely on the identification of relevant variables that are specific to the system under study. As a result, they fail to address the generality of collective motion as an emergent behaviour across systems. In this paper, we show how topological data analysis allows abstracting collective motion patterns and defining meaningful distances between them. The proposed method uses persistent homology to represent the topological features of velocity fields as sets of persistence diagrams. We show how a distance between these sets can discriminate the various behaviours exhibited by a system, and we demonstrate its generality by applying it to a fish school, a pedestrian crowd, and a robot swarm. We validate our results against those obtained through system-specific abstractions, and illustrate how to use topological representations to explain discrepancies. In short, we show how to quantify and interpret differences between collective motion patterns for any system for which a velocity field can be constructed. This will allow researchers in complex systems to adopt a common framework for collective motion analysis, enabling direct comparison of emergent behaviours across disciplines.