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These network systems are typically spatially distributed over a large area and may consists of hundreds of agents in smart-sensor networks to millions of agents in social networks. As such, they do not possess a central coordinator or a central point for access to the complete system information. This lack of central entity makes the traditional (centralized) optimization and control techniques inapplicable, thus necessitating the development of new distributed computational models and algorithms to support efficient operations over such networks. This tutorial provides an overview of the convergence rate of distributed algorithms for coordination and its relevance to optimization in a system of autonomous agents embedded in a communication network, where each agent is aware of (and can communicate with) its local neighbors only. The focus is on distributed averaging dynamics for consensus problems and its role in consensusbased gradient methods for convex optimization problems, where the network objective function is separable across the constituent agents.<\/jats:p>","DOI":"10.1561\/2600000004","type":"journal-article","created":{"date-parts":[[2015,6,11]],"date-time":"2015-06-11T08:45:26Z","timestamp":1434012326000},"page":"1-100","source":"Crossref","is-referenced-by-count":72,"title":["Convergence Rate of Distributed Averaging Dynamics and Optimization in Networks"],"prefix":"10.1561","volume":"2","author":[{"given":"Angelia","family":"Nedi\u0107","sequence":"first","affiliation":[{"name":"University of Illinois at Urbana-Champaign"}]}],"member":"140","published-online":{"date-parts":[[2015,6,11]]},"reference":[{"issue":"3","key":"2026033114224199400_ref001","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","article-title":"Theory of reproducing 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self-driven particles","volume":"75","author":"Vicsek","year":"1995","journal-title":"Physical Review Letters"},{"key":"2026033114224199400_ref194","volume-title":"Functional Operators","author":"von Neumann","year":"1950"},{"key":"2026033114224199400_ref195","first-page":"49","article-title":"Event-triggered distributed optimization in sensor networks","author":"Wan","year":"2009","journal-title":"Symposium on Information Processing of Sensor Networks, (San Francisco, CA)"},{"key":"2026033114224199400_ref196","first-page":"3800","article-title":"A control perspective for centralized and distributed convex optimization","author":"Wang","year":"2011","journal-title":"Proceedings of the IEEE Conference on Decision and Control, (Florida, USA)"},{"key":"2026033114224199400_ref197","first-page":"5445","article-title":"Distributed alternating direction method of multipliers","author":"Wei","year":"2012","journal-title":"Proceedings of the 51st IEEE Conference on Decision and Control and European Control Conference"},{"key":"2026033114224199400_ref198","first-page":"551","article-title":"On the O(1\/k) convergence of asynchronous distributed alternating direction method of multipliers","author":"Wei","year":"2013","journal-title":"Proceedings of IEEE Global Conference on Signal and Information Processing"},{"key":"2026033114224199400_ref199","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.sysconle.2004.02.022","article-title":"Fast linear iterations for distributed averaging","volume":"53","author":"Xiao","year":"2004","journal-title":"Systems and Control Letters"},{"issue":"3","key":"2026033114224199400_ref200","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/s10957-006-9080-1","article-title":"Optimal scaling of a gradient method for distributed resource allocation","volume":"129","author":"Xiao","year":"2006","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026033114224199400_ref201","first-page":"63","article-title":"A scheme for robust distributed sensor fusion based on average consensus","author":"Xiao","year":"2005","journal-title":"Proceedings of the 4th International Conference on Information Processing in Sensor Networks"},{"issue":"1","key":"2026033114224199400_ref202","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.jpdc.2006.08.010","article-title":"Distributed average consensus with least mean square deviation","volume":"67","author":"Xiao","year":"2007","journal-title":"Journal of Parallel and Distributed Computing"},{"issue":"11","key":"2026033114224199400_ref203","doi-asserted-by":"crossref","first-page":"2483","DOI":"10.1109\/TKDE.2012.191","article-title":"Distributed autonomous online learning: Regrets and intrinsic privacy-preserving properties","volume":"25","author":"Yan","year":"2013","journal-title":"IEEE Transactions on Data and Knowledge 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