On the Impact of Random Actions on Opinion Dynamics

TL;DR

This paper analyzes opinion dynamics in social networks with stubborn agents, demonstrating convergence to a stubborn agent’s opinion under various models, including random and drifting agent scenarios, and critiques recent related literature.

Contribution

It extends the understanding of opinion convergence by weakening assumptions in the DeGroot model and analyzing random and drifting agent influences.

Findings

Consensus is achieved even with weaker assumptions.

Random Bernoulli opinion models converge to the stubborn agent’s opinion.

Herding occurs with a drifting agent as well.

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. We also consider a variation on this model where the stubborn agent is replaced with a drifting agent and show that herding is achieved also in this case. Finally, we offer a detailed critique of a proof regarding a claim about a closely related model in the recent literature.