On Lyapunov functions and gradient flow structures in linear consensus systems

TL;DR

This paper explores Lyapunov functions and gradient flow structures in linear consensus systems, unifying various energy and information measures within a geometric framework to analyze stability and convergence.

Contribution

It introduces a class of additive convex Lyapunov functions and establishes a general gradient formalism for both linear and non-linear consensus dynamics.

Findings

Weighted convex functions serve as Lyapunov functions for consensus systems.

Gradient structures can be generalized from Euclidean to Riemannian geometries.

Information-theoretic Lyapunov functions relate to consensus dynamics.

Abstract

Quadratic Lyapunov functions are prevalent in stability analysis of linear consensus systems. In this paper we show that weighted sums of convex functions of the different coordinates are Lyapunov functions for irreducible consensus systems. We introduce a particular class of additive convex Lyapunov functions that includes as particular instances the stored electric energy in RC circuits, Kullback-Leiber's information divergence between two probability measures, and Gibbs free energy in chemical reaction networks. On that basis we establish a general gradient formalism that allows to represent linear symmetric consensus dynamics as a gradient descent dynamics of any additive convex Lyapunov function, for a suitable choice of local scalar product. This result generalizes the well-known Euclidean gradient structures of consensus to general Riemannian ones. We also find natural non-linear consensus dynamics differing only from linearity by a different Riemannian structure, sharing a same Lyapunov function. We also see how information-theoretic Lyapunov functions, common in Markov chain theory, generate linear consensus through an appropriate gradient descent. From this unified perspective we hope to open avenues for the study of convergence for linear and non-linear consensus alike.