Hegselmann-Krause and Cucker-Smale type models with attractive-repulsive interaction

TL;DR

This paper studies opinion formation and flocking models with alternating attractive and repulsive interactions, proving convergence to consensus and flocking under general conditions, with some results showing exponential convergence.

Contribution

It introduces analysis of Hegselmann-Krause and Cucker-Smale models with attractive-repulsive interactions, establishing convergence and flocking results under broad assumptions.

Findings

Proved convergence to consensus in Hegselmann-Krause model with positive-negative interactions.

Established asymptotic flocking in Cucker-Smale model with mixed interactions.

Demonstrated exponential convergence under additional conditions.

Abstract

In this paper, we analyze a Hegselmann-Krause opinion formation model and a Cucker-Smale flocking model with attractive-repulsive interaction. To be precise, we investigate the situation in which the individuals involved in an opinion formation or a flocking process attract each other in certain time intervals and repeal each other in other ones. Under quite general assumptions, we prove the convergence to consensus for the Hegselmann-Krause model and the exhibition of asymptotic flocking for the Cucker-Smale model in presence of positive-negative interaction. With some additional conditions, we are able to improve the convergence to consensus for the solutions of the Hegselmann-Krause model, namely we establish an exponential convergence to consensus result.