Opinion Dynamics with Random Actions and a Stubborn Agent

TL;DR

This paper analyzes opinion dynamics in social networks with stubborn agents, showing that under certain models, opinions converge to that of the stubborn agent, even with randomness and weaker assumptions.

Contribution

It extends existing models by proving convergence to the stubborn agent's opinion under weaker conditions and with stochastic opinion updates.

Findings

Consensus is achieved in the DeGroot model with stubborn agents.

Random Bernoulli opinion models also converge to the stubborn agent's opinion.

Convergence occurs in probability, not necessarily almost surely.

Abstract

We study opinion dynamics in a social network with stubborn agents who influence their neighbors but who themselves always stick to their initial opinion. We consider first the well-known DeGroot model. While it is known in the literature that this model can lead to consensus even in the presence of a stubborn agent, we show that the same result holds under weaker assumptions than has been previously reported. We then consider a recent extension of the DeGroot model in which the opinion of each agent is a random Bernoulli distributed variable, and by leveraging on the first result we establish that this model also leads to consensus, in the sense of convergence in probability, in the presence of a stubborn agent. Moreover, all agents' opinions converge to that of the stubborn agent.