{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:59Z","timestamp":1787323259265,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"8","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We study a simple continuous-time multiagent system related to Krause's model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multiagent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a nontrivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.<\/jats:p>","DOI":"10.1137\/090766188","type":"journal-article","created":{"date-parts":[[2010,10,28]],"date-time":"2010-10-28T18:09:04Z","timestamp":1288289344000},"page":"5214-5240","source":"Crossref","is-referenced-by-count":173,"title":["Continuous-Time Average-Preserving Opinion Dynamics with Opinion-Dependent Communications"],"prefix":"10.1137","volume":"48","author":[{"given":"Vincent D.","family":"Blondel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julien M.","family":"Hendrickx","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John N.","family":"Tsitsiklis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,10,28]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis,\n                      Convergence in multiagent coordination, consensus, and flocking\n                      , in Proceedings of the 44th IEEE Conference on Decision and Control (CDC'2005), Seville, Spain, 2005, pp. 2996\u20133000.","DOI":"10.1109\/CDC.2005.1582620"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"V. D. Blondel, J. M. Hendrickx, and J. N. Tsitsiklis,\n                      On the 2R conjecture for multi-agent systems\n                      , in Proceedings of the European Control Conference (ECC'2007), Kos, Greece, 2007, pp. 874\u2013881.","DOI":"10.23919\/ECC.2007.7068885"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2009.2031211"},{"key":"R4","unstructured":"V. D. Blondel, J. M. Hendrickx, and J. N. Tsitsiklis,\n                      Existence and Uniqueness of Solutions for a Continuous-Time Opinion Dynamics Model with State-Dependent Connectivity\n                      , http:\/\/web.mit.edu\/jnt\/www\/Papers\/BHT10-solutions-DA.pdf (2009)."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"C. Canuto, F. Fagnani, and P. Tilli,\n                      A Eulerian approach to the analysis of rendez-vous algorithms\n                      , in Proceedings of the 17th IFAC World Congress (IFAC'2008), 2008, pp. 9039\u20139044.","DOI":"10.3182\/20080706-5-KR-1001.01526"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.895842"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s11537-007-0647-x"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729991"},{"key":"R9","unstructured":"J. M. Hendrickx,\n                      Graphs and Networks for the Analysis of Autonomous Agent Systems\n                      , Ph.D. thesis, Universit\u00e9 catholique de Louvain, Louvain-la-Neuve, Belgium; available online from http:\/\/www.inma.ucl.ac.be\/%7Ehendrickx\/availablepublications\/Thesis_Julien_Hendrickx.pdf (2008)."},{"key":"R10","unstructured":"J. M. Hendrickx and V. D. Blondel,\n                      Convergence of different linear and non-linear Vicsek models\n                      , in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS'2006), Kyoto, Japan, 2006, pp. 1229\u20131240."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2003.812781"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2003.10.004"},{"key":"R13","unstructured":"U. Krause,\n                      Soziale Dynamiken mit vielen Interakteuren. Eine Problemskizze\n                      , in Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren, Bremen University, Bremen, Germany, 1997, pp. 37\u201351."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2005.02.086"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1142\/S0129183107011789"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2004.841888"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2006.887293"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1109\/MCS.2007.338264"},{"key":"R19","unstructured":"J. N. Tsitsiklis,\n                      Problems in Decentralized Decision Making and Computation\n                      , Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA; available online from http:\/\/web.mit.edu\/jnt\/www\/PhD-84-jnt.pdf (1984)."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.75.1226"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090766188","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:51:26Z","timestamp":1787320286000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090766188"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":20,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090766188"],"URL":"https:\/\/doi.org\/10.1137\/090766188","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}