  Collective motion in animal groups, such as swarms of insects, flocks of
birds, and schools of fish, are some of the most visually striking examples of
emergent behavior. Empirical analysis of these behaviors in experiment or
computational simulation primarily involves the use of "swarm-averaged" metrics
or order parameters such as velocity alignment and angular momentum. Recently,
tools from computational topology have been applied to the analysis of swarms
to further understand and automate the detection of fundamentally different
swarm structures evolving in space and time. Here, we show how the field of
graph signal processing can be used to fuse these two approaches by
collectively analyzing swarm properties using graph Fourier harmonics that
respect the topological structure of the swarm. This graph Fourier analysis
reveals hidden structure in a number of common swarming states and forms the
basis of a flexible analysis framework for collective motion.
