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.

In the field of graph signal processing (GSP), directed graphs present a particular challenge for the "standard approaches" of GSP to due to their asymmetric nature. The presence of negative-or complex-weight directed edges, a graphical structure used in fields such as neuroscience, critical infrastructure, and robot coordination, further complicates the issue. Recent results generalized the total variation of a graph signal to that of directed variation as a motivating principle for developing a graphical Fourier transform (GFT). Here, we extend these techniques to concepts of signal variation appropriate for indefinite and complex-valued graphs and use them to define a GFT for these classes of graph. Simulation results on random graphs are presented, as well as a case study of a portion of the fruit fly connectome.

Collective motion among biological organisms such as insects, fish, and birds has motivated considerable interest not only in biology but also in distributed robotic systems. In a robotic or biological swarm, anomalous agents (whether malfunctioning or nefarious) behave differently than the normal agents and attempt to hide in the "chaos" of the swarm. By defining a graph structure between agents in a swarm, we can treat the agents' properties as a graph signal and use tools from the field of graph signal processing to understand local and global swarm properties. Here, we leverage this idea to show that anomalous agents can be effectively detected using their impacts on the graph Fourier structure of the swarm.

Graph signal processing (GSP) is an emerging field developed for analyzing signals defined on irregular spatial structures modeled as graphs. Given the considerable literature regarding the resilience of infrastructure networks using graph theory, it is not surprising that a number of applications of GSP can be found in the resilience domain. GSP techniques assume that the choice of graphical Fourier transform (GFT) imparts a particular spectral structure on the signal of interest. We assess a number of power distribution systems with respect to metrics of signal structure and identify several correlates to system properties and further demonstrate how these metrics relate to performance of some GSP techniques. We also discuss the feasibility of a data-driven approach that improves these metrics and apply it to a water distribution scenario. Overall, we find that many of the candidate systems analyzed are properly structured in the chosen GFT basis and amenable to GSP techniques, but identify considerable variability and nuance that merits future investigation.