Products of Generalized Stochastic Sarymsakov Matrices

TL;DR

This paper extends the class of stochastic matrices known as Sarymsakov matrices by introducing a generalized concept that includes larger subsets, providing new conditions for convergence to rank-one matrices relevant to consensus algorithms.

Contribution

The paper introduces a generalized framework for Sarymsakov matrices using an SIA index, enlarging the class of matrices with convergence properties, and offers new sufficient conditions for product convergence.

Findings

Larger classes of matrices with convergence properties identified

Sufficient conditions for convergence to rank-one matrices established

Enhanced understanding of consensus algorithms derived

Abstract

In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of stochastic Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication and (ii) the infinitely long left-product of the elements from a compact subset converges to a rank-one matrix. In this paper, we show that a larger subset with these two properties can be derived by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved either by introducing an "SIA index", whose value is one for Sarymsakov matrices, and then looking at those stochastic matrices with larger SIA indices, or by considering matrices that are not even SIA. Besides constructing a larger set, we give sufficient conditions for generalized Sarymsakov matrices so that their products converge to rank-one matrices. The new insight gained through studying generalized Sarymsakov matrices and their products has led to a new understanding of the existing results on consensus algorithms and will be helpful for the design of network coordination algorithms.