Laplacian Spectra of Two-Layer Hierarchical Cyclic Pursuit Schemes

Cyclic pursuit is one of the oldest multi-agent strategies with many interesting features. The vast majority of the papers dedicated to this strategy cover various extensions related to the models of interacting agents, delays, uncertainties, asynchronous communication, etc. A certain line of research studies hierarchical topologies that extend the conventional single-layer scheme. Our paper contributes to this line. Motivated by the fact that such structures are scalable, we study the spectral properties of their Laplacian matrices. First, we consider a two-layer cyclic pursuit strategy and analyze its Laplacian spectrum as the number of agents tends to infinity. Next, we propose a more sparse two-layer topology, study its spectrum, and describe the curves that contain a limit location of the eigenvalues of the corresponding Laplacian matrix.