A Laplacian Approach to Stubborn Agents and their Role in Opinion Formation on Influence Networks

TL;DR

This paper introduces a Laplacian-based analytical model to understand how stubborn agents influence opinion formation and diversity in social networks, revealing effects of network structure on consensus and heterogeneity.

Contribution

It develops a novel Laplacian approach to analytically characterize the impact of stubborn agents on opinion dynamics in networked systems.

Findings

Stubborn agents significantly affect opinion consensus and heterogeneity.

Small-world networks reduce opinion diversity and enhance coherence.

Community structure influences opinion diversity and the ease of consensus change.

Abstract

Within the framework of a simple model for social influence, the Taylor model, we analytically investigate the role of stubborn agents in the overall opinion dynamics of networked systems. Similar to zealots, stubborn agents are biased towards a certain opinion and have a major effect on the collective opinion formation process. Based on a modified version of the network Laplacian we derive quantities capturing the transient dynamics of the system and the emerging stationary opinion states. In the case of a single stubborn agent we characterize his/her ability to coherently change a prevailing consensus. For two antagonistic stubborn agents we investigate the opinion heterogeneity of the emerging non-consensus states and describe their statistical properties using a graph metric similar to the resistance distance in electrical networks. Applying the model to synthetic and empirical networks we find while opinion diversity is decreased by small-worldness and favored in the case of a pronounced community structure the opposite is true for the coherence of opinions during a consensus change.