# Kinetic models for topological nearest-neighbor interactions

## TL;DR

This paper develops kinetic models for agents with topological nearest-neighbor interactions, deriving a spatial diffusion equation as the system size grows and connecting it to smooth rank-based models.

## Contribution

It introduces a new kinetic framework for topological interactions and links it to existing smooth rank-based models, expanding understanding of agent-based systems.

## Key findings

- Kinetic equations are derived for systems with topological interactions.
- The models converge to a spatial diffusion equation in the large system limit.
- The same kinetic equation applies when agents interact with their K closest neighbors.

## Abstract

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal andhuman behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times arandomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinityand, 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 correspondingkinetic 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.