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Kinetic models for topological nearest-neighbor interactions

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  • Blanchet, Adrien
  • Degond, Pierre

Abstract

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in [10]. The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

Suggested Citation

  • Blanchet, Adrien & Degond, Pierre, 2017. "Kinetic models for topological nearest-neighbor interactions," TSE Working Papers 17-786, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:31577
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    References listed on IDEAS

    as
    1. Anders Eriksson & Martin Nilsson Jacobi & Johan Nyström & Kolbjørn Tunstrøm, 2010. "Determining interaction rules in animal swarms," Behavioral Ecology, International Society for Behavioral Ecology, vol. 21(5), pages 1106-1111.
    2. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
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    Keywords

    rank-based interaction; spatial diffusion equation; continuity equation; concentration of measure;
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