Hierarchical Cyclic Pursuit: Algebraic Curves Containing the Laplacian Spectra

This article addresses the problem of multiagent communication in networks with a regular directed ring structure. These can be viewed as hierarchical extensions of the classical cyclic pursuit topology. We show that the spectra of the corresponding Laplacian matrices allow exact localization on the complex plane. Furthermore, we derive a general form of the characteristic polynomial of such matrices, analyze the algebraic curves its roots belong to, and propose a way to obtain their closed-form equations. In combination with frequency-domain consensus criteria for high-order single-input single-output linear agents, these curves enable one to analyze the feasibility of consensus in networks with a varying number of agents.