Hierarchical Cyclic Pursuit: Algebraic Curves Containing the Laplacian Spectra

TL;DR

This paper explores the spectral properties of Laplacian matrices in hierarchical cyclic pursuit networks, deriving algebraic curves and characteristic polynomials to analyze consensus feasibility in multi-agent systems.

Contribution

It introduces a novel algebraic and geometric framework for understanding Laplacian spectra in hierarchical cyclic pursuit networks, enabling precise analysis of consensus conditions.

Findings

Laplacian spectra can be localized exactly on the complex plane.

A general form of the characteristic polynomial for these matrices is derived.

Algebraic curves containing the roots are characterized and their equations obtained.

Abstract

The paper addresses the problem of multi-agent communication in networks with 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 SISO linear agents, these curves enable one to analyze the feasibility of consensus in networks with varying number of agents.