Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions

TL;DR

This paper introduces a topological interaction-based flocking model inspired by starling behavior, develops new analytical tools for its analysis, and derives a mean-field limit leading to continuum descriptions, revealing novel collective dynamics.

Contribution

It presents a new topological flocking model, develops analytical methods for its study, and derives a mean-field limit with a novel concept of relative separation.

Findings

Conditions for asymptotic flocking are established.

A rigorous derivation of the mean-field limit is provided.

Numerical simulations show new pattern dynamics in the modified model.

Abstract

We introduce a Cucker-Smale-type model for flocking, where the strength of interaction between two agents depends on their relative separation (called "topological distance" in previous works), which is the number of intermediate individuals separating them. This makes the model scale-free and is motivated by recent extensive observations of starling flocks, suggesting that interaction ruling animal collective behavior depends on topological rather than metric distance. We study the conditions leading to asymptotic flocking in the topological model, defined as the convergence of the agents' velocities to a common vector. The shift from metric to topological interactions requires development of new analytical methods, taking into account the graph-theoretical nature of the problem. Moreover, we provide a rigorous derivation of the mean-field limit of large populations, recovering kinetic and hydrodynamic descriptions. In particular, we introduce the novel concept of relative separation in continuum descriptions, which is applicable to a broad variety of models of collective behavior. As an example, we shortly discuss a topological modification of the attraction-repulsion model and illustrate with numerical simulations that the modified model produces interesting new pattern dynamics.