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 Blanchet and Degond (J Stat Phys
163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as well
as some concentration of measure arguments.