Document Zbl 1235.93014

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Then a fundamental question is: how many such leaders are needed? Naturally the answer depends on the model to be studied. In this paper a typical model proposed by Vicsek et al. is used for answering the question. By estimating the characteristics concerning the initial states of all agents and analyzing the system dynamics, we provide lower bounds on the ratio of leaders needed to guarantee the expected consensus.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6011964">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93A14" title="MSC2020">93A14</a> </td> <td class="space"> Decentralized systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aproportion">proportion</a>; <a href="/?q=ut%3Aleader-follower+model">leader-follower model</a>; <a href="/?q=ut%3Aconsensus">consensus</a>; <a href="/?q=ut%3Amulti-agent+systems">multi-agent systems</a>; <a href="/?q=ut%3AVicsek%27s+model">Vicsek’s model</a></div> <div class="software"> <h3>Software:</h3><a href="/software/38863">Boids</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1235.93014" data-ciurl="/ci/06011964" data-biburl="/bibtex/06011964.bib" data-amsurl="/amsrefs/06011964.bib" data-xmlurl="/xml/06011964.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06011964.pdf" title="Zbl 1235.93014 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.automatica.2011.08.047" aria-label="DOI for “The proportion of leaders needed for the expected consensus”" title="10.1016/j.automatica.2011.08.047">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Chow, Y. S.; Teicher, H., Probability theory: independence, interchangeability, martingales (1978), Springer-Verlag: Springer-Verlag New York · <a href="/0399.60001" class="nowrap">Zbl 0399.60001</a></td> </tr><tr> <td>[2]</td> <td class="space">Couzin, I. D.; Krause, J.; Franks, N. R.; Levy, S., Effective leadership and decision-making in animal groups on the move, Nature, 433, 513-516 (2005)</td> </tr><tr> <td>[3]</td> <td class="space">Czirok, A.; Jacob, E. B.; Cohen, I.; Schochet, O.; Vicsek, T., Formation of complex bacterial colonies via self-generated vortices, Physics Review E, 54, 2, 1791-1801 (1996)</td> </tr><tr> <td>[4]</td> <td class="space">Halloy, J.; Sempo, G.; Caprari, G.; Rivault, C.; Asadpour, M.; Tâche, F., Social integration of robots into croups of cockroaches to control self-organized choices, Science, 318, 1155-1158 (2007)</td> </tr><tr> <td>[5]</td> <td class="space">Han, J.; Li, M.; Guo, L., Soft control on collective behavior of a group of autonomous agents by a shill agent, Journal of Systems Science and Complexity, 19, 1, 54-62 (2006)</td> </tr><tr> <td>[6]</td> <td class="space">Helbing, D.; Farkas, I.; Vicsek, T., (Simulating dynamic features of the escape panic. Simulating dynamic features of the escape panic, Nature, Vol. 407 (2000), Springer-Verlag: Springer-Verlag New York), 487-490</td> </tr><tr> <td>[7]</td> <td class="space">Jadbabaie, A.; Lin, J.; Morse, A. S., Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions Automation Control, 48, 9, 988-1001 (2003) · <a href="/1364.93514" class="nowrap">Zbl 1364.93514</a></td> </tr><tr> <td>[8]</td> <td class="space">Liu, Z. X. (2008). How many leaders are required for consensus? In <span class="zbmathjax-textit">Proc. of the 27th chinese control conference</span>; Liu, Z. X. (2008). How many leaders are required for consensus? In <span class="zbmathjax-textit">Proc. of the 27th chinese control conference</span></td> </tr><tr> <td>[9]</td> <td class="space">Liu, Z. X.; Guo, L., Synchronization of multi-agent systems without connectivity assumptions, Automatica, 45, 12, 2744-2753 (2009) · <a href="/1192.93057" class="nowrap">Zbl 1192.93057</a></td> </tr><tr> <td>[10]</td> <td class="space">Liu, Z. X., Han, J., & Hu, X. (2009). The number of leaders needed for consensus. In <span class="zbmathjax-textit">Proc. of the 48th IEEE conf. decision and control</span>; Liu, Z. X., Han, J., & Hu, X. (2009). The number of leaders needed for consensus. In <span class="zbmathjax-textit">Proc. of the 48th IEEE conf. decision and control</span></td> </tr><tr> <td>[11]</td> <td class="space">Nabet, N. B.; Leonard, N. E.; Couzin, I. D.; Levy, S. A., Dynamics of decision-making in animal group motion, Journal of Nonlinear Science, 19, 399-435 (2009) · <a href="/1169.93302" class="nowrap">Zbl 1169.93302</a></td> </tr><tr> <td>[12]</td> <td class="space">Olfati-Saber, R., Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Transactions Automation Control, 51, 3, 401-420 (2006) · <a href="/1366.93391" class="nowrap">Zbl 1366.93391</a></td> </tr><tr> <td>[13]</td> <td class="space">Reynolds, C., Flocks, herds, and schools: a distributed behavioral model, Computer Graphics, 21, 4, 25-34 (1987)</td> </tr><tr> <td>[14]</td> <td class="space">Sumpter, D. J.T., The principles of collective animal behaviour, Philosophical Transactions of the Royal Society of London. Series B, 361, 1465, 5-22 (2006) · <a href="/1120.92032" class="nowrap">Zbl 1120.92032</a></td> </tr><tr> <td>[15]</td> <td class="space">Tang, G. G.; Guo, L., Convergence of a class of multi-agent systems in probabilistic frameworks, Journal of Systems Science and Complexity, 20, 2, 173-197 (2007) · <a href="/1130.68088" class="nowrap">Zbl 1130.68088</a></td> </tr><tr> <td>[16]</td> <td class="space">Vicsek, T.; Czirók, A.; Ben-Jacob, E.; Cohen, I.; Sochet, O., Novel type of phase transition in a system of self-driven particles, Physics Review Letters, 75, 1226-1229 (1995)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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