On final opinions of the Friedkin-Johnsen model over random graphs with partially stubborn community

TL;DR

This paper analyzes how the final opinions in the Friedkin-Johnsen model on random graphs with community structures concentrate around those of an expected graph model, highlighting the influence of stubbornness and network connectivity.

Contribution

It provides probabilistic bounds for the opinion differences in the Friedkin-Johnsen model on random graphs with community structures, extending understanding of opinion dynamics in such networks.

Findings

Final opinions concentrate around expected graph model as network size grows.

Link probability between stubborn and non-stubborn agents significantly affects opinion distance.

Higher stubbornness reduces the opinion difference when all agents are stubborn.

Abstract

This paper studies the formation of final opinions for the Friedkin-Johnsen (FJ) model with a community of partially stubborn agents. The underlying network of the FJ model is symmetric and generated from a random graph model, in which each link is added independently from a Bernoulli distribution. It is shown that the final opinions of the FJ model will concentrate around those of an FJ model over the expected graph as the network size grows, on the condition that the stubborn agents are well connected to other agents. Probability bounds are proposed for the distance between these two final opinion vectors, respectively for the cases where there exist non-stubborn agents or not. Numerical experiments are provided to illustrate the theoretical findings. The simulation shows that, in presence of non-stubborn agents, the link probability between the stubborn and the non-stubborn communities affect the distance between the two final opinion vectors significantly. Additionally, if all agents are stubborn, the opinion distance decreases with the agent stubbornness.