Consensus Seminorms and their Applications

TL;DR

This paper explores the use of seminorms to analyze and bound the convergence rate in consensus problems, correcting previous misconceptions and extending known results to broader classes of matrices.

Contribution

It revisits and corrects earlier seminorms used in consensus analysis and introduces a wider family of seminorms that guarantee exponential convergence rates.

Findings

Corrected a previous error regarding a seminorm's relation to the coefficient of ergodicity.

Introduced a broader family of seminorms ensuring exponential convergence.

Showed limitations of seminorms in bounding convergence for larger matrix classes.

Abstract

Consensus is a well-studied problem in distributed sensing, computation and control, yet deriving useful and easily computable bounds on the rate of convergence to consensus remains a challenge. This paper discusses the use of seminorms for this goal. A previously suggested family of seminorms is revisited, and an error made in their original presentation is corrected, where it was claimed that the a certain seminorm is equal to the well-known coefficient of ergodicity. Next, a wider family of seminorms is introduced, and it is shown that contraction in any of these seminorms guarantees convergence at an exponential rate of infinite products of matrices, generalizing known results on stochastic matrices to the class of matrices whose row sums are all equal one. Finally, it is shown that such seminorms cannot be used to bound the rate of convergence of classes larger than the well-known class of scrambling matrices.