Order preservation in a generalized version of Krause's opinion dynamics model

TL;DR

This paper investigates when the order of opinions remains unchanged in a generalized opinion dynamics model, providing a necessary and sufficient condition on the influence function for order preservation.

Contribution

It introduces a generic version of Krause's model with an influence function and characterizes the exact conditions for order preservation, addressing variations in the model.

Findings

Derived a necessary and sufficient condition for order preservation.

Identified that some natural variations of the model are not order preserving.

Provided insights into the theoretical analysis of opinion dynamics models.

Abstract

Krause's model of opinion dynamics has recently been the object of several studies, partly because it is one of the simplest multi-agent systems involving position-dependent changing topologies. In this model, agents have an opinion represented by a real number and they update it by averaging those agent opinions distant from their opinion by less than a certain interaction radius. Some results obtained on this model rely on the fact that the opinion orders remain unchanged under iteration, a property that is consistent with the intuition in models with simultaneous updating on a fully connected communication topology. Several variations of this model have been proposed. We show that some natural variations are not order preserving and therefore cause potential problems with the theoretical analysis and the consistence with the intuition. We consider a generic version of Krause's model parameterized by an influence function that encapsulates most of the variations proposed in the literature. We then derive a necessary and sufficient condition on this function for the opinion order to be preserved.