Analyzing Collective Motion Using Graph Fourier Analysis

Abstract

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.

Detecting Anomalous Swarming Agents with Graph Signal Processing

“…The necessary ingredients for performing GSP analysis are the specification of a graph and a function defined on the vertices of that graph. For a collection of N swarming entities with positions x j and velocities v j both ∈ R n (time indices surpressed), following [10] we define each agent as a vertex in a graph, and for each instance in time we construct an ad-…”

“…In a natural definition for non-negative weighted symmetric graphs, the GFT uses the eigenvectors of L as basis functions instead of the complex exponentials used in the Fourier transform. However, [10] demonstrated that L may be better suited for analysis of swarms. Using the eigendecomposition L = UΛU , with U a unitary matrix of eigenvectors, and Λ a diagonal matrix of eigenvalues in increasing order, the GFT of a graph signal f is defined as f = U f and the corresponding inverse GFT as f = U f .…”

“…This topological approach was further extended in [10] to analyze how the local and global "order" of swarm properties varies with respect to the graph structure. The analysis in [10] employed the field of graph signal processing (GSP) and graph Fourier analysis to show that common swarm states were highly structured (i.e., band-limited) when viewed in the graph Fourier domain.…”

“…This topological approach was further extended in [10] to analyze how the local and global "order" of swarm properties varies with respect to the graph structure. The analysis in [10] employed the field of graph signal processing (GSP) and graph Fourier analysis to show that common swarm states were highly structured (i.e., band-limited) when viewed in the graph Fourier domain. Broadly speaking, GSP builds on its roots in spectral graph theory [11] and algebraic signal processing [12] to generalize concepts from classical signal pro-cessing to signals defined on the vertices of irregular domains modeled by graphs [13][14][15][16].…”

“…In this work, we show how the GFT structure of swarms revealed in [10] can be used to design "graph filters" that operate on the swarm state to detect agents within the swarm whose dynamics (and thus behavior) are fundamentally different from the bulk of the swarm. These anomalous agents have behaviors that differ in subtle ways from the rest of the swarm; from a purely kinematic perspective the anomalous trajectories are consistent with the non-anomalous agents.…”