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Friedkin-Johnsen Model with Diminishing Competition

Luca Ballotta, Áron Vékássy, Stephanie Gil, Michal Yemini

TL;DR

This work extends the Friedkin-Johnsen framework by introducing a diminishing, uniform competition parameter $\lambda_t$ in the update $x_{t+1} = (1-\lambda_t)W x_t + \lambda_t x_0$, bridging DeGroot consensus and classic FJ dynamics. It proves that, as $\lambda_t\to0$, the system converges to the nominal consensus value $x_{ss} = (v^\top x_0)\mathbf{1}$, and provides explicit upper and lower bounds on the convergence rate that depend on the second-largest singular value $\sigma_{\text{max}}$ of $W$ and the decay of $\lambda_t$; in particular, a slow decay like $\lambda_t=1/(t+1)$ yields $\rho(t)=O(1/t)$. A negative result shows that non-uniform competition across agents can prevent convergence to the nominal consensus, highlighting a limitation for fully decentralized designs. Numerical experiments on random graphs validate the theoretical findings, demonstrating nominal consensus under uniform decaying competition and deviation under non-uniform competition. The results have implications for resilient and safe decentralized coordination where agents gradually identify adversaries while preserving a nominal agreement.

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.

Friedkin-Johnsen Model with Diminishing Competition

TL;DR

This work extends the Friedkin-Johnsen framework by introducing a diminishing, uniform competition parameter in the update , bridging DeGroot consensus and classic FJ dynamics. It proves that, as , the system converges to the nominal consensus value , and provides explicit upper and lower bounds on the convergence rate that depend on the second-largest singular value of and the decay of ; in particular, a slow decay like yields . A negative result shows that non-uniform competition across agents can prevent convergence to the nominal consensus, highlighting a limitation for fully decentralized designs. Numerical experiments on random graphs validate the theoretical findings, demonstrating nominal consensus under uniform decaying competition and deviation under non-uniform competition. The results have implications for resilient and safe decentralized coordination where agents gradually identify adversaries while preserving a nominal agreement.

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.
Paper Structure (16 sections, 6 theorems, 44 equations, 1 figure)

This paper contains 16 sections, 6 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup
  3. Notation and Mathematical Preliminaries
  4. System Model
  5. FJ Model With Uniform Time-Varying Competition
  6. Convergence Analysis
  7. Convergence to Consensus
  8. Convergence Rate
  9. Non-Uniform Competition: A Negative Result
  10. Numerical Results
  11. Conclusion
  12. Proof of \ref{['lemma:convergence-input']}
  13. Proof of \ref{['thm:consensus']}
  14. Proof of \ref{['prop:convergence-rate-upper-bound']}
  15. Proof of \ref{['prop:convergence-rate-lower-bound']}
  16. ...and 1 more sections

Key Result

Lemma 1

If $\lim_{t\rightarrow\infty}\lambda_t=0$, then there exists $y\in { {\mathbb R}^{N} }$ such that

Figures (1)

  • Figure 1: Average logarithmic distance of $20$ agents from the nominal consensus value using the three different competition parameters $\lambda_t$ described in \ref{['sec:sims']}.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 1: Protocol \ref{['eq:fj']} achieves the nominal consensus
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2: Convergence rate of \ref{['eq:fj']}
  • proof
  • ...and 4 more