Bounded confidence dynamics and graph control: enforcing consensus

TL;DR

This paper introduces a novel control mechanism for bounded confidence models that guarantees unconditional convergence to consensus by maintaining network connectivity, supported by rigorous proofs and numerical evidence.

Contribution

It proposes the No one left behind dynamics, a new control method ensuring consensus in bounded confidence models, with a relaxed version preserving key features and convergence.

Findings

Unconditional convergence to consensus is achieved with the proposed control.

The dynamics preserve network connectivity throughout the evolution.

Numerical evidence suggests sharp convergence rates are difficult to quantify.

Abstract

A generic feature of bounded confidence type models is the formation of clusters of agents. We propose and study a variant of bounded confidence dynamics with the goal of inducing unconditional convergence to a consensus. The defining feature of these dynamics, which we name the No one left behind dynamics, is the introduction of a local control on the agents which preserves the connectivity of the interaction network. We rigorously demonstrate that these dynamics result in unconditional convergence to a consensus. The qualitative nature of our argument prevents us quantifying how fast a consensus emerges, however we present numerical evidence that sharp convergence rates would be challenging to obtain for such dynamics. Finally, we propose a relaxed version of the control. The dynamics that result maintain many of the qualitative features of the bounded confidence dynamics yet ultimately still converge to a consensus as the control still maintains connectivity of the interaction network.