[Paper Review] Opinion Dynamics with Stubborn Agents.

David Scott Hunter, Tauhid Zaman|arXiv (Cornell University)|Jun 29, 2018

Opinion Dynamics and Social Influence24 references8 citations

TL;DR

This paper proposes a dynamic opinion model with time-varying stubbornness, showing that opinions converge to a linear equilibrium under general conditions. It introduces harmonic influence centrality and formulates stubborn agent placement as a monotone submodular optimization problem, enabling effective influence with few agents via greedy algorithms.

ABSTRACT

We consider the problem of optimizing the placement of stubborn agents in a social network in order to maximally impact population opinions. We assume individuals in a directed social network each have a latent opinion that evolves over time in response to social media posts by their neighbors. The individuals randomly communicate noisy versions of their latent opinion to their neighbors. Each individual updates his opinion using a time-varying update rule that has him become more stubborn with time and be less affected by new posts. The dynamic update rule is a novel component of our model and reflects realistic behaviors observed in many psychological studies. We show that in the presence of stubborn agents with immutable opinions and under fairly general conditions on the stubbornness rate of the individuals, the opinions converge to an equilibrium determined by a linear system. We give an interesting electrical network interpretation of the equilibrium. We also use this equilibrium to present a simple closed form expression for harmonic influence centrality, which is a function that quantifies how much a node can affect the mean opinion in a network. We develop a discrete optimization formulation for the problem of maximally shifting opinions in a network by targeting nodes with stubborn agents. We show that this is an optimization problem with a monotone and submodular objective, allowing us to utilize a greedy algorithm. Finally, we show that a small number of stubborn agents can non-trivially influence a large population using simulated networks.

Motivation & Objective

• To model how opinions evolve in a directed social network with time-varying individual stubbornness.
• To understand how immutable stubborn agents impact the equilibrium opinion distribution.
• To develop a tractable optimization framework for placing stubborn agents to maximize opinion shift.
• To characterize the equilibrium opinion state using an electrical network analogy.
• To derive a closed-form expression for harmonic influence centrality as a measure of node influence.

Proposed method

• Model individuals as having latent opinions that update via noisy, time-varying communication with neighbors.
• Introduce a time-varying update rule where individuals become more stubborn over time, reducing sensitivity to new information.
• Prove convergence to a linear equilibrium system under general conditions on stubbornness rates.
• Establish an electrical network interpretation of the equilibrium, linking network structure to opinion outcomes.
• Formulate the stubborn agent placement problem as a discrete optimization task with a monotone and submodular objective function.
• Apply a greedy algorithm to efficiently solve the optimization problem due to submodularity.

Experimental results

Research questions

• RQ1 How does time-varying stubbornness affect opinion convergence in a social network?
• RQ2 What is the equilibrium opinion distribution when stubborn agents with fixed opinions are introduced?
• RQ3 Can the equilibrium be interpreted through an electrical network analogy?
• RQ4 How can harmonic influence centrality be derived in closed form for nodes in the network?
• RQ5 What is the optimal placement strategy for stubborn agents to maximize opinion shift, and how effective is it?

Key findings

• Opinions converge to a linear equilibrium under general conditions on individual stubbornness rates, even with noisy communication.
• The equilibrium state admits an electrical network interpretation, linking network topology to opinion outcomes.
• A closed-form expression for harmonic influence centrality is derived, quantifying a node's potential to affect mean opinion.
• The opinion shift optimization problem is monotone and submodular, enabling efficient solution via a greedy algorithm.
• Simulations demonstrate that a small number of strategically placed stubborn agents can significantly influence the overall population opinion.