Sparse control of Hegselmann-Krause models: Black hole and declustering

TL;DR

This paper develops control strategies to prevent opinion clustering in Hegselmann-Krause models by introducing an entropy-based measure and analyzing conditions for declustering in finite and kinetic systems.

Contribution

It proposes a novel entropy-type functional for declustering control and characterizes conditions to avoid clustering, including black holes and safety zones.

Findings

Entropy functional effectively measures dispersion.

Conditions for declustering depend on initial data.

Identification of black holes and safety zones.

Abstract

This paper elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).