Abstract
The collective behavior of multi-agent systems is an important studying point for the investigation of complex systems, and a basic model of multi-agent systems is the so called Vicsek model, which possesses some key features of complex systems, such as dynamic behavior, local interaction, changing neighborhood, etc. This model looks simple, but the nonlinearly coupled relationship makes the theoretical analysis quite complicated. Jadbabaie et al. analyzed the linearized heading equations in this model and showed that all agents will synchronize eventually, provided that the neighbor graphs associated with the agents’ positions satisfy a certain connectivity condition. Much subsequent research effort has been devoted to the analysis of the Vicsek model since the publication of Jadbabaie’s work. However, an unresolved key problem is when such a connectivity is satisfied. This paper given a sufficient condition to guarantee the synchronization of the Vicsek model, which is imposed on the model parameters only. Moreover, some counterexamples are given to show that the connectivity of the neighbor graphs is not sufficient for synchronization of the Vicsek model if the initial headings are allowed to be in [0, 2π), which reveals some fundamental differences between the Vicsek model and its linearized version.
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References
Shaw E. Fish in schools. Nat Hist, 1975, 84(8): 40–46
Reynolds C. Flocks, herds, and schools: a distributed behavioral model. Comp Graph, 1987, 21(4): 25–34
Vicsek T, Czirok A, Ben-Jacob E, et al. Novel type of phase transition in a system of self-driven particles. Phys Rev Lett, 1995, 75(6): 1226–1229
Jadbabaie A, Lin J, Morse A S. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Contr, 2003, 48(6): 988–1001
Cucker F, Smale S. Emergent behavior in flocks. IEEE Trans Autom Contr, 2007, 52(5): 852–862
Savkin A V. Coordinated collective motion of groups of autonomous mobile robots: analysis of Vicsek model. IEEE Trans Autom Contr, 2004, 39(6): 981–983
Tang G G, Guo L. Convergence of a class of multi-agent systems in probabilistic framework. J Syst Sci Compl, 2007, 20(2): 173–197
Liu Z X, Guo L. Connectivity and synchronization of multi-agent systems. In: Proc 25th Chinese Control Conference (in Chinese), Hrbin, 2006, 373–378
Hendrickx J M, Blondel V D. Convergence of different linear and non-linear Vicsek models. In: Proc. 17th Int. Symp. on MTNS. 2006. 24–28
Wolfowitz J. Products of indecomposable aperiodic stochastic matrices. Proc Amer Math Soc, 1963, 14(5): 733–737
Liu Z X. Collective behavior of multi-agent systems with local rules (in Chinese). PhD Thesis. Beijing: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2007
Han J, Li M, Guo L. Soft control on collective behavior of a group of autonomous agents by a shill agent. J Syst Sci Compl, 2006, 19(1): 54–62
Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985
Wang L, Liu Z X, Guo L. Robust consensus of multi-agent systems with noise. In: Proc. the 26th Chinese Control Conference, Zhangjiajie, 2007. 737–741
Liu Z X, Guo L. Synchronization of Vicsek model with large population. In: Proc. the 26th Chinese Control Conference, Zhangjiajie, 2007. 673–677
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Supported by the National Natural Science Foundation of China (Grant Nos. 60221301 and 60334040)
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Liu, Z., Guo, L. Connectivity and synchronization of Vicsek model. Sci. China Ser. F-Inf. Sci. 51, 848–858 (2008). https://doi.org/10.1007/s11432-008-0077-2
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DOI: https://doi.org/10.1007/s11432-008-0077-2