Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation

Haskovec, Jan

doi:10.1090/proc/15522

Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation
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Proc. Amer. Math. Soc. 149 (2021), 3425-3437 Request permission

Abstract:

We consider a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed $\mathfrak {c}>0$. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than $\mathfrak {c}$. We also provide sufficient conditions for asymptotic consensus in the spatially multidimensional setting.

References

Arnab Bhattacharyya, Mark Braverman, Bernard Chazelle, and Huy L. Nguyễn, On the convergence of the Hegselmann-Krause system, ITCS’13—Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science, ACM, New York, 2013, pp. 61–65. MR 3385386
Vincent D. Blondel, Julien M. Hendrickx, and John N. Tsitsiklis, On Krause’s multi-agent consensus model with state-dependent connectivity, IEEE Trans. Automat. Control 54 (2009), no. 11, 2586–2597. MR 2571922, DOI 10.1109/TAC.2009.2031211
Scott Camazine, Jean-Louis Deneubourg, Nigel R. Franks, James Sneyd, Guy Theraulaz, and Eric Bonabeau, Self-organization in biological systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003. Reprint of the 2001 original. MR 2343706
C. Canuto, F. Fagnani, and P. Tilli, An Eulerian approach to the analysis of Krause’s consensus models, SIAM J. Control Optim. 50 (2012), no. 1, 243–265. MR 2888264, DOI 10.1137/100793177
Adrián Carro, Raúl Toral, and Maxi San Miguel, The role of noise and initial conditions in the asymptotic solution of a bounded confidence, continuous-opinion model, J. Stat. Phys. 151 (2013), no. 1-2, 131–149. MR 3045810, DOI 10.1007/s10955-012-0635-2
C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics. Rev. Mod. Phys. 81 (2009), 591–646.
Y.-P. Choi, A. Paolucci, and C. Pignotti, Consensus of the Hegselmann-Krause opinion formation model with time delay. Math. Methods Appl. Sci. (2020), 1– 20.
H. Hamman, Swarm robotics: a formal approach. Springer, 2018.
R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simul. 5 (2002), 1–24.
Pierre-Emmanuel Jabin and Sebastien Motsch, Clustering and asymptotic behavior in opinion formation, J. Differential Equations 257 (2014), no. 11, 4165–4187. MR 3264419, DOI 10.1016/j.jde.2014.08.005
Ali Jadbabaie, Jie Lin, and A. Stephen Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control 48 (2003), no. 6, 988–1001. MR 1986266, DOI 10.1109/TAC.2003.812781
P. Krugman, The self organizing economy. Blackwell Publishers, 1995.
J. Lu, D. W. C. Ho, and J. Kurths, Consensus over directed static networks with arbitrary finite communications delays. Phys. Rev. E 80 (2009), 066121.
S. Mohajer and B. Touri, On convergence rate of scalar Hegselmann-Krause dynamics. Proceedings of the IEEE American Control Conference (ACC) (2013).
Luc Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Trans. Automat. Control 50 (2005), no. 2, 169–182. MR 2116423, DOI 10.1109/TAC.2004.841888
Sebastien Motsch and Eitan Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev. 56 (2014), no. 4, 577–621. MR 3274797, DOI 10.1137/120901866
Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani (eds.), Mathematical modeling of collective behavior in socio-economic and life sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Ltd., Boston, MA, 2010. MR 2761862, DOI 10.1007/978-0-8176-4946-3
Hal Smith, An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, vol. 57, Springer, New York, 2011. MR 2724792, DOI 10.1007/978-1-4419-7646-8
Gabriele Valentini, Achieving consensus in robot swarms, Studies in Computational Intelligence, vol. 706, Springer, Cham, 2017. Design and analysis of strategies for the best-of-$n$ problem; With a foreword by Marco Dorigo. MR 3679622, DOI 10.1007/978-3-319-53609-5
T. Vicsek and A. Zafeiris, Collective motion. Phys. Rep. 517 (2012), 71–140.
Chu Wang, Qianxiao Li, Weinan E, and Bernard Chazelle, Noisy Hegselmann-Krause systems: phase transition and the $2R$-conjecture, J. Stat. Phys. 166 (2017), no. 5, 1209–1225. MR 3610211, DOI 10.1007/s10955-017-1718-x
Edvin Wedin and Peter Hegarty, A quadratic lower bound for the convergence rate in the one-dimensional Hegselmann-Krause bounded confidence dynamics, Discrete Comput. Geom. 53 (2015), no. 2, 478–486. MR 3316234, DOI 10.1007/s00454-014-9657-7
H. Xu, H. Wang, and Z. Xuan, Opinion dynamics: a multidisciplinary review and perspective on future research. Int. J. Knowl. Syst. Sci. 2 (2011), 72–91.
C. Williams, The \textdollar{}300m cable that will save traders milliseconds. The Telegraph, 11 Sep 2011. https://www.telegraph.co.uk/technology/news/8753784/The-300m-cable-that-will-save-traders-milliseconds.html