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 . The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.
rank-based interaction; spatial diffusion equation; continuity equation; concentration of measure;
Adrien Blanchet, and Pierre Degond, “Kinetic models for topological nearest-neighbor interactions”, TSE Working Paper, n. 17-786, March 2017.
Adrien Blanchet, and Pierre Degond, “Kinetic models for topological nearest-neighbor interactions”, Journal of Statistical Physics, vol. 169, n. 5, December 2017, pp. 929–950.
Journal of Statistical Physics, vol. 169, n. 5, December 2017, pp. 929–950