The Deformed Consensus Protocol: Extended Version

TL;DR

This paper introduces a generalized consensus protocol using a deformed Laplacian, analyzing its stability across various graph types and parameters, supported by theoretical insights and simulations.

Contribution

It presents a novel deformed Laplacian-based consensus protocol and analyzes its stability for different graph structures and parameter values.

Findings

Stability depends on the parameter 's' and graph topology.

The deformed consensus protocol generalizes the standard one.

Simulation results confirm theoretical predictions.

Abstract

This paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable 's' which reduces to the standard Laplacian for 's' equal to unity. The stability properties of the ensuing deformed consensus protocol are studied in terms of parameter 's' for some special families of undirected and directed graphs, and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems. Examples and simulation results are provided to illustrate our theoretical findings.