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On the minimum number of neighbors needed for consensus of flocks

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Abstract

This paper investigates consensus of flocks consisting of n autonomous agents in the plane, where each agent has the same constant moving speed v n and updates its heading by the average value of the k n nearest agents from it, with v n and k n being two prescribed parameters depending on n. Such a topological interaction rule is referred to as k n -nearest-neighbors rule, which has been validated for a class of birds by biologists and verified to be robust with respect to disturbances. A theoretical analysis will be presented for this flocking model under a random framework with large population, but without imposing any a priori connectivity assumptions. We will show that the minimum number of k n needed for consensus is of the order O(log n) in a certain sense. To be precise, there exist two constants C1 > C2 > 0 such that, if k n > C1 log n, then the flocking model will achieve consensus for any initial headings with high probability, provided that the speed v n is suitably small. On the other hand, if k n < C2 log n, then for large n, with probability 1, there exist some initial headings such that consensus cannot be achieved, regardless of the value of v n .

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Authors and Affiliations

  1. Noah’s Ark Laboratory, 2012 Lab, Huawei Technologies Co. Ltd, Beijing, 100085, China

    Chen Chen

  2. LSC & NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ge Chen & Lei Guo

Authors
  1. Chen Chen
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  2. Ge Chen
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  3. Lei Guo
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Corresponding author

Correspondence to Lei Guo.

Additional information

This paper is dedicated to Professor T. J. Tarn on the occasion of his 80th birthday.

This work was supported by the National Natural Science Foundation of China (No. 91427304, 61673373, 11688101), the National Key Basic Research Program of China (973 program) (No. 2014CB845301/2/3), and the Leading Research Projects of Chinese Academy of Sciences (No. QYZDJ-SSW-JSC003).

Chen CHEN was born in Shannxi, China, in 1987. She received the B.Sc. degree in Mathematics from Beihang University in 2009, and the Ph.D. degree in Control Theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, in 2014. She is currently a researcher at Huawei Technologies Co. Ltd. Her research interests include complex systems, distributed filters, reinforcement learning and deep learning.

Ge CHEN received the B.Sc. degree in Mathematics from the University of Science and Technology of China in 2004, and the Ph.D. degree in Mathematics from the University of Chinese Academy of Sciences, China, in 2009. He jointed the National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2011, and is currently an Associate Professor. His current research interest is the collective behavior of multi-agent systems. Dr. Chen received the First Prize of the Application Award from the Operations Research Society of China (2010). One of his papers was selected as a SIGEST paper by the SIAM Review (2014). He was also a finalist for the OR in Development prize from the International Federation of Operations Research Societies (2011), and for the best theoretical paper award at the 10th World Congress on Intelligent Control and Automation (2012).

Lei GUO was born in China in 1961. He received the B.Sc. degree in Mathematics from Shandong University in 1982, and the Ph.D. degree in Control Theory from the Chinese Academy of Sciences (CAS) in 1987. He is currently a professor at the Academy of Mathematics and Systems Science, CAS, and Director of National Center for Mathematics and Interdisciplinary Sciences (NCMIS), CAS. He is a Fellow of both IEEE and IFAC, Member of CAS, Foreign Member of the Royal Swedish Academy of Engineering Sciences. His current research interests include feedback capability, nonlinear and adaptive control, multi-agent systems, distributed adaptive filtering, and game-based control systems, among others.

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Chen, C., Chen, G. & Guo, L. On the minimum number of neighbors needed for consensus of flocks. Control Theory Technol. 15, 327–339 (2017). https://doi.org/10.1007/s11768-017-7097-7

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  • DOI: https://doi.org/10.1007/s11768-017-7097-7

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