On backward product of stochastic matrices

Abstract

We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.

Convergence of max–min consensus algorithms

“…Various necessary and/or sufficient conditions are established for the proposed max-min consensus algorithm under time-dependent interaction graphs. These conditions are consistent with the infinite flow property and persistent connectivity conditions in the literature which are utilized to study consensus algorithms (Hendrickx & Tsitsiklis, 2013;Martin & Girard, 2013;Touri & Nedic, 2011, 2012. The derived convergence conditions for directed graphs do not rely on the condition on the positive lower bound of the arc weights, which usually show up for the study of standard consensus algorithms.…”

Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

“…Yet, for a sequence of doubly-stochastic matrices, it is sufficient to assume that the diagonal entries are bounded away from 0. This result was proved by Touri and Nedić (theorem 5 of [16] or theorem 7 of [15], relying on theorem 6 of [17]). We provide a simpler proof and a slight improvement, showing that the sequence (M n .…”

On indigenous random consensus and averaging dynamics

“…As defined in [30], in the deterministic setting, a sequence {S(k)} of subsets of [m] is called regular if |S(k)| = |S(0)| ≥ 1 for all k, i.e. the cardinality of S(k) does not change with time.…”

“…Many of the works in this field have been focused on the study of deterministic averaging dynamics [9], [10], [11], [12], [13], [14], [15]. To address practical issues such as link-failure and random disturbances in the links, there has been an increasing interest in the study of random averaging dynamics [16], [17], [18], [19], [20], [21], [22], [23].…”