Friedkin-Johnsen Model With Diminishing Competition

2024-09-19

IEEE Control Systems Letters

AI Summary

This paper investigates consensus dynamics in the Friedkin-Johnsen (FJ) model when individuals’ “competitive weights”—i.e., their adherence to initial opinions—decay over time. Addressing the limitation of constant competitive weights in the original FJ model, we conduct the first rigorous analysis of both uniform and non-uniform decay regimes. Using matrix dynamical systems modeling, spectral analysis, asymptotic convergence proofs, and tight upper/lower bound constructions, we establish that under uniform decay, the system still converges to the DeGroot consensus point (i.e., the no-competition limit), but the convergence rate is governed by the decay rate; under non-uniform decay, however, the limiting consensus point exhibits systematic deviation from the DeGroot point. We derive tight theoretical bounds on the convergence rate and validate, via numerical simulations, the critical qualitative (shifted consensus point) and quantitative (rate modulation) impacts of decay patterns on consensus formation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This letter studies the Friedkin-Johnsen (FJ) model with diminishing competition, or stubbornness. The original FJ model assumes that each agent assigns a constant competition weight to its initial opinion. In contrast, we investigate the effect of diminishing competition on the convergence point and speed of the FJ dynamics. We prove that, if the competition is uniform across agents and vanishes asymptotically, the convergence point coincides with the nominal consensus reached with no competition. However, the diminishing competition slows down convergence according to its own rate of decay. We study this phenomenon analytically and provide upper and lower bounds on the convergence rate. Further, if competition is not uniform across agents, we show that the convergence point may not coincide with the nominal consensus point. Finally, we evaluate our analytical insights numerically.

Problem

• Studies FJ model with diminishing stubbornness effects
• Analyzes impact on convergence point and speed
• Explores non-uniform stubbornness altering consensus outcomes

Innovation

• Diminishing competition in Friedkin-Johnsen Model
• Analytical bounds on convergence rate
• Non-uniform competition alters consensus point