Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks

In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over gener...
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Autor*in:	Basar, Tamer [verfasserIn] Etesami, Seyed Rasoul

Format:	Artikel
Sprache:	Englisch

Erschienen:	2016

Schlagwörter:	Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis

Übergeordnetes Werk:	Enthalten in: IEEE transactions on automatic control - New York, NY : Inst., 1963, 61(2016), 2, Seite 443-455
Übergeordnetes Werk:	volume:61 ; year:2016 ; number:2 ; pages:443-455

Links:	Volltext Link aufrufen

DOI / URN:	10.1109/TAC.2015.2440568

Katalog-ID:	OLC1970418907

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245	1	0	\|a Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
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520			\|a In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs.
650		4	\|a Robot sensing systems
650		4	\|a Protocols
650		4	\|a Heuristic algorithms
650		4	\|a convergence time
650		4	\|a Upper bound
650		4	\|a random walk
650		4	\|a Harmonic analysis
650		4	\|a Markov chains
650		4	\|a Convergence
650		4	\|a spectral representation
650		4	\|a Quantized consensus
650		4	\|a time varying networks
650		4	\|a Markov processes
650		4	\|a Probability
650		4	\|a Graphs
650		4	\|a Markov analysis
700	1		\|a Etesami, Seyed Rasoul \|4 oth
773	0	8	\|i Enthalten in \|t IEEE transactions on automatic control \|d New York, NY : Inst., 1963 \|g 61(2016), 2, Seite 443-455 \|w (DE-627)129601705 \|w (DE-600)241443-0 \|w (DE-576)015095320 \|x 0018-9286 \|7 nnns
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allfields	10.1109/TAC.2015.2440568 doi PQ20160307 (DE-627)OLC1970418907 (DE-599)GBVOLC1970418907 (PRQ)c990-ff77629e3897655fca91a153e79a2a8b98c67a48b9c8626391b17be5bdebe50f0 (KEY)0005057120160000061000200443convergencetimeforunbiasedquantizedconsensusoverst DE-627 ger DE-627 rakwb eng 620 DNB Basar, Tamer verfasserin aut Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis Etesami, Seyed Rasoul oth Enthalten in IEEE transactions on automatic control New York, NY : Inst., 1963 61(2016), 2, Seite 443-455 (DE-627)129601705 (DE-600)241443-0 (DE-576)015095320 0018-9286 nnns volume:61 year:2016 number:2 pages:443-455 http://dx.doi.org/10.1109/TAC.2015.2440568 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_11 GBV_ILN_30 GBV_ILN_70 GBV_ILN_120 GBV_ILN_193 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2333 GBV_ILN_4193 AR 61 2016 2 443-455
spelling	10.1109/TAC.2015.2440568 doi PQ20160307 (DE-627)OLC1970418907 (DE-599)GBVOLC1970418907 (PRQ)c990-ff77629e3897655fca91a153e79a2a8b98c67a48b9c8626391b17be5bdebe50f0 (KEY)0005057120160000061000200443convergencetimeforunbiasedquantizedconsensusoverst DE-627 ger DE-627 rakwb eng 620 DNB Basar, Tamer verfasserin aut Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis Etesami, Seyed Rasoul oth Enthalten in IEEE transactions on automatic control New York, NY : Inst., 1963 61(2016), 2, Seite 443-455 (DE-627)129601705 (DE-600)241443-0 (DE-576)015095320 0018-9286 nnns volume:61 year:2016 number:2 pages:443-455 http://dx.doi.org/10.1109/TAC.2015.2440568 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_11 GBV_ILN_30 GBV_ILN_70 GBV_ILN_120 GBV_ILN_193 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2333 GBV_ILN_4193 AR 61 2016 2 443-455
allfields_unstemmed	10.1109/TAC.2015.2440568 doi PQ20160307 (DE-627)OLC1970418907 (DE-599)GBVOLC1970418907 (PRQ)c990-ff77629e3897655fca91a153e79a2a8b98c67a48b9c8626391b17be5bdebe50f0 (KEY)0005057120160000061000200443convergencetimeforunbiasedquantizedconsensusoverst DE-627 ger DE-627 rakwb eng 620 DNB Basar, Tamer verfasserin aut Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis Etesami, Seyed Rasoul oth Enthalten in IEEE transactions on automatic control New York, NY : Inst., 1963 61(2016), 2, Seite 443-455 (DE-627)129601705 (DE-600)241443-0 (DE-576)015095320 0018-9286 nnns volume:61 year:2016 number:2 pages:443-455 http://dx.doi.org/10.1109/TAC.2015.2440568 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_11 GBV_ILN_30 GBV_ILN_70 GBV_ILN_120 GBV_ILN_193 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2333 GBV_ILN_4193 AR 61 2016 2 443-455
allfieldsGer	10.1109/TAC.2015.2440568 doi PQ20160307 (DE-627)OLC1970418907 (DE-599)GBVOLC1970418907 (PRQ)c990-ff77629e3897655fca91a153e79a2a8b98c67a48b9c8626391b17be5bdebe50f0 (KEY)0005057120160000061000200443convergencetimeforunbiasedquantizedconsensusoverst DE-627 ger DE-627 rakwb eng 620 DNB Basar, Tamer verfasserin aut Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis Etesami, Seyed Rasoul oth Enthalten in IEEE transactions on automatic control New York, NY : Inst., 1963 61(2016), 2, Seite 443-455 (DE-627)129601705 (DE-600)241443-0 (DE-576)015095320 0018-9286 nnns volume:61 year:2016 number:2 pages:443-455 http://dx.doi.org/10.1109/TAC.2015.2440568 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_11 GBV_ILN_30 GBV_ILN_70 GBV_ILN_120 GBV_ILN_193 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2333 GBV_ILN_4193 AR 61 2016 2 443-455
allfieldsSound	10.1109/TAC.2015.2440568 doi PQ20160307 (DE-627)OLC1970418907 (DE-599)GBVOLC1970418907 (PRQ)c990-ff77629e3897655fca91a153e79a2a8b98c67a48b9c8626391b17be5bdebe50f0 (KEY)0005057120160000061000200443convergencetimeforunbiasedquantizedconsensusoverst DE-627 ger DE-627 rakwb eng 620 DNB Basar, Tamer verfasserin aut Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis Etesami, Seyed Rasoul oth Enthalten in IEEE transactions on automatic control New York, NY : Inst., 1963 61(2016), 2, Seite 443-455 (DE-627)129601705 (DE-600)241443-0 (DE-576)015095320 0018-9286 nnns volume:61 year:2016 number:2 pages:443-455 http://dx.doi.org/10.1109/TAC.2015.2440568 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT GBV_ILN_11 GBV_ILN_30 GBV_ILN_70 GBV_ILN_120 GBV_ILN_193 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2333 GBV_ILN_4193 AR 61 2016 2 443-455
language	English
source	Enthalten in IEEE transactions on automatic control 61(2016), 2, Seite 443-455 volume:61 year:2016 number:2 pages:443-455
sourceStr	Enthalten in IEEE transactions on automatic control 61(2016), 2, Seite 443-455 volume:61 year:2016 number:2 pages:443-455
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis
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The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. 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author	Basar, Tamer
spellingShingle	Basar, Tamer ddc 620 misc Robot sensing systems misc Protocols misc Heuristic algorithms misc convergence time misc Upper bound misc random walk misc Harmonic analysis misc Markov chains misc Convergence misc spectral representation misc Quantized consensus misc time varying networks misc Markov processes misc Probability misc Graphs misc Markov analysis Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
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topic_title	620 DNB Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks Robot sensing systems Protocols Heuristic algorithms convergence time Upper bound random walk Harmonic analysis Markov chains Convergence spectral representation Quantized consensus time varying networks Markov processes Probability Graphs Markov analysis
topic	ddc 620 misc Robot sensing systems misc Protocols misc Heuristic algorithms misc convergence time misc Upper bound misc random walk misc Harmonic analysis misc Markov chains misc Convergence misc spectral representation misc Quantized consensus misc time varying networks misc Markov processes misc Probability misc Graphs misc Markov analysis
topic_unstemmed	ddc 620 misc Robot sensing systems misc Protocols misc Heuristic algorithms misc convergence time misc Upper bound misc random walk misc Harmonic analysis misc Markov chains misc Convergence misc spectral representation misc Quantized consensus misc time varying networks misc Markov processes misc Probability misc Graphs misc Markov analysis
topic_browse	ddc 620 misc Robot sensing systems misc Protocols misc Heuristic algorithms misc convergence time misc Upper bound misc random walk misc Harmonic analysis misc Markov chains misc Convergence misc spectral representation misc Quantized consensus misc time varying networks misc Markov processes misc Probability misc Graphs misc Markov analysis
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title	Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
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title_full	Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
author_sort	Basar, Tamer
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title_sort	convergence time for unbiased quantized consensus over static and dynamic networks
title_auth	Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
abstract	In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs.
abstractGer	In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs.
abstract_unstemmed	In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of <inline-formula> <tex-math notation="TeX">O(nmD\log n)</tex-math></inline-formula> for the expected convergence time to consensus, where <inline-formula> <tex-math notation="TeX">n</tex-math></inline-formula>, <inline-formula> <tex-math notation="TeX">m</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">D</tex-math></inline-formula>, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of <inline-formula> <tex-math notation="TeX">\log n</tex-math></inline-formula> for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs.
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title_short	Convergence Time for Unbiased Quantized Consensus Over Static and Dynamic Networks
url	http://dx.doi.org/10.1109/TAC.2015.2440568 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7117385
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up_date	2024-07-03T15:17:03.784Z
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