Hierarchical Cyclic Pursuit: Algebraic Curves Containing the Laplacian Spectra

TL;DR

The paper addresses the localization of Laplacian spectra for hierarchical ring digraphs (ring digraphs with macro-vertices) by showing the spectra lie on high-order algebraic curves of order $2n$, with a general polynomial form $\Delta(\lambda)=(P_n(\lambda))^{m}-(-1)^N$ and an algorithm to obtain the implicit curve $f(x,y)=0$. It classifies rings, derives curve equations for several cases (e.g., Cassini ovals for $n=2$ and sextics for $n=3$), and extends to weighted rings; these spectral loci enable a frequency-domain consensus analysis via the region $\Omega$ associated with the transfer function $\phi(s)=a(s)/b(s)$, yielding consensus criteria that are invariant to the number of agents $N$. The results provide geometric, algebraic, and algorithmic tools for assessing consensus feasibility in scalable, high-order multi-agent networks on ring topologies. This work offers practical insights for designing robust, scalable consensus protocols in hierarchical cyclic pursuit and related networked control systems.

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.

Figures (20)

• Figure 1: A Hamiltonian cycle corresponding to the cyclic pursuit strategy with four agents (a) and a macro-vertex (b)
• Figure 2: A ring digraph with four macro-vertices
• Figure 3: A macro-vertex (a) on four nodes can be obtained by connecting two macro-vertices of type (b) by a directed arc.
• Figure 4: Two simple rings ((a) and (b)) and a complex ring (c) constructed as the round replication of the simple ring (b)
• Figure 5: The quantity $Y(N)$ as function of the number of nodes

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Lemma 1
• Remark 2
• Definition 6
• Theorem 1