%0 Journal Article
%@ 1572-9613
%A Blanchet, Adrien
%A Degond, Pierre
%D 2017
%F publications:25738
%I Kluwer
%J Journal of Statistical Physics
%K rank-based interaction
%K spatial diffusion equation
%K continuity equation
%K concentration of measure
%N 5
%P 929-950
%R 10.1007/s10955-017-1882-z
%T Kinetic models for topological nearest-neighbor interactions
%U https://publications.ut-capitole.fr/id/eprint/25738/
%V 169
%X 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.