Consensus for the Hegselmann-Krause model with time variable time delays

TL;DR

This paper proves that the Hegselmann-Krause opinion model with variable and distributed time delays always converges exponentially to consensus, regardless of the delay size, provided the influence function remains positive.

Contribution

It establishes exponential convergence to consensus in Hegselmann-Krause models with general time delays without smallness restrictions, extending previous results.

Findings

Exponential convergence to consensus under positive influence functions.

No smallness assumption needed on time delays.

Extension to models with distributed time delays.

Abstract

In this paper, we analyze a Hegselmann-Krause opinion formation model with time-variable time delay and prove that, if the influence function is always positive, then there is exponential convergence to consensus without requiring any smallness assumptions on the time delay function. The analysis is then extended to a model with distributed time delay.