Lyapunov Approach to Consensus Problems

TL;DR

This paper introduces a Lyapunov-based analysis for consensus algorithms, deriving new convergence rates that incorporate network structure, applicable to both constrained and unconstrained scenarios with general weight matrices.

Contribution

It develops a Lyapunov approach for analyzing consensus dynamics, extending convergence results to constrained cases with non-doubly stochastic weights.

Findings

Exponential convergence rate for unconstrained consensus with rate of order 1 - O(1/(m log m))

Exponential convergence for constrained consensus with general weight matrices

Graph structure influences convergence speed in the proposed analysis

Abstract

This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of $1-O(1/(m\log_2m))$. Also, the exponential convergence rate is established for constrained consensus, which extends the existing results limited to the use of doubly stochastic weight matrices.