Contractions for consensus processes

Liu, J.; Morse, A. Stephen; Anderson, Brian D. O.; Yu, Changbin

doi:10.1109/cdc.2011.6160989

Cited by 18 publications

(10 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, m}. From (37) and Lemma 1, it follows that S i ∩ S i = ∅, S k = S r for 1 ≤ k < r ≤ 3 n − 2 n+1 + 2. Without loss of generality, assume that k = 1.…”

Section: Appendix Ementioning

confidence: 89%

“…Even in the light of these classical as well as recent findings, the following fundamental question remains: What is the largest subset of the class of n × n stochastic matrices whose compact subsets are all consensus sets? In [37], this question is answered under the assumption that each stochastic matrix has positive diagonal entries. For general stochastic matrices, however, the question has remained open.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalized Sarymsakov Matrices

Xia

Liu

Cao

et al. 2019

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication, and more critically (ii) whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved by introducing the notion of the SIA index of a stochastic matrix, whose value is 1 for Sarymsakov matrices, and then exploring those stochastic matrices with larger SIA indices. In addition to constructing the larger set, the paper introduces another class of generalized Sarymsakov matrices, which contains matrices that are not SIA, and studies their products. Sufficient conditions are provided for an infinite product of matrices from this class, converging to a rank-one matrix. Finally, as an application of the results just described and to confirm their usefulness, a necessary and sufficient combinatorial condition, the "avoiding set condition", for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited. In addition, a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set.

show abstract

“…, m}. From (37) and Lemma 1, it follows that S i ∩ S i = ∅, S k = S r for 1 ≤ k < r ≤ 3 n − 2 n+1 + 2. Without loss of generality, assume that k = 1.…”

Section: Appendix Ementioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Generalized Sarymsakov Matrices

Xia

Liu

Cao

et al. 2019

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

show abstract

“…There are several different ways to prove this theorem using ideas from [2], [5], [6], [13], [14]. A proof of this theorem will be given in the full length version of this paper.…”

Section: A Gossiping Sequencesmentioning

confidence: 97%

Request-based gossiping

Liu

Mou

Morse

et al. 2011

IEEE Conference on Decision and Control and European Control Conference

Self Cite

View full text Add to dashboard Cite

By the distributed averaging problem is meant the problem of computing the average value of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. Gossiping is a well-known approach to the problem which seeks to iteratively arrive at a solution by allowing each agent to interchange information with at most one neighbor at each iterative step. Crafting a gossiping protocol which accomplishes this is challenging because gossiping is an inherently collaborative process which can lead to deadlock unless careful precautions are taken to ensure that it does not. In this paper we present three gossiping protocols. We show by example that the first can deadlock. While the second cannot, it requires a degree of network-wide coordination which may not be possible to secure in some applications. The third protocol uses only local information, is guaranteed to avoid deadlock, and requires fewer transmissions per iteration than standard broadcastbased distributed averaging protocols.

show abstract

“…We've also sought to understand the subtleties of aperiodic gossiping [14]. In [15] we show that the classical concept of a contraction coefficient which is important in quantifying convergence rates for nonhomogeneous Markov Chains, could be viewed as but one of a number of different types of semi-norms of suitably defined stochastic matrices. More directly relevant to gossiping, in [12] we've used simple examples to show that the largely ignored problem of devising a gossiping protocol which avoids deadlock while generating rapidly converging gossiping sequences is especially formidable [14].…”

mentioning

confidence: 99%

“…6. We invented new tools for analyzing convergence rates for consensus and gossiping algorithms [7,12,15].…”

mentioning

confidence: 99%

Intelligent Distributed Control

Morse¹

2012

Self Cite

View full text Add to dashboard Cite

Standard Form 298 (Rev. 8/98) REPORT DOCUMENTATION PAGE Prescribed by ANSI Std. Z39.18Form The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to the Department of Defense, Executive Services and Communications Directorate (0704-0188). Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

show abstract

Contractions for consensus processes

Cited by 18 publications

References 27 publications

Generalized Sarymsakov Matrices

Generalized Sarymsakov Matrices

Request-based gossiping

Intelligent Distributed Control

Contact Info

Product

Resources

About