How friends and non-determinism affect opinion dynamics

TL;DR

This paper explores how social relationships and non-deterministic movements influence opinion formation models, showing that modified systems still converge efficiently despite their differences from the classical model.

Contribution

It introduces and analyzes two variants of the HK system incorporating social networks and noise, proving their polynomial convergence.

Findings

Both variants converge in polynomial time.

Social network constraints do not hinder convergence.

Noise in opinion updates still allows for efficient convergence.

Abstract

The Hegselmann-Krause system (HK system for short) is one of the most popular models for the dynamics of opinion formation in multiagent systems. Agents are modeled as points in opinion space, and at every time step, each agent moves to the mass center of all the agents within unit distance. The rate of convergence of HK systems has been the subject of several recent works. In this work, we investigate two natural variations of the HK system and their effect on the dynamics. In the first variation, we only allow pairs of agents who are friends in an underlying social network to communicate with each other. In the second variation, agents may not move exactly to the mass center but somewhere close to it. The dynamics of both variants are qualitatively very different from that of the classical HK system. Nevertheless, we prove that both these systems converge in polynomial number of non-trivial steps, regardless of the social network in the first variant and noise patterns in the second variant.