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Coalescing Force of Group Pressure: Consensus in Nonlinear Opinion Dynamics

Iryna Zabarianska, Anton V. Proskurnikov

TL;DR

This work addresses how group pressure drives consensus in nonlinear, time-varying opinion dynamics with multidimensional opinions and general, opinion-dependent weights. It introduces the ODGP framework, combining an underlying averaging process with a public-opinion term and proving convergence under mild conditions: asymptotic consensus if the minimal conformity $p_*(t)$ satisfies $\sum_{t=0}^\infty p_*(t)=\infty$, exponential convergence when $p_*(t)\ge p_0>0$, and finite-time consensus if some $p_*(t_0)=1$. The key contributions include generalizing prior results to arbitrary underlying dynamics and public opinions within the convex hull, deriving a convergence-time bound independent of group size $n$ for positive conformity, and providing HK-specific corollaries supported by numerical simulations. The findings offer robust theoretical guarantees for consensus under diverse social-influence scenarios and have practical implications for interpreting polls and online network dynamics.

Abstract

This work extends the recent opinion dynamics model from Cheng et al., emphasizing the role of group pressure in consensus formation. We generalize the findings to incorporate social influence algorithms with general time-varying, opinion-dependent weights and multidimensional opinions, beyond bounded confidence dynamics. We demonstrate that, with uniformly positive conformity levels, group pressure consistently drives consensus and provide a tighter estimate for the convergence rate. Unlike previous models, the common public opinion in our framework can assume arbitrary forms within the convex hull of current opinions, offering flexibility applicable to real-world scenarios such as opinion polls with random participant selection. This analysis provides deeper insights into how group pressure mechanisms foster consensus under diverse conditions.

Coalescing Force of Group Pressure: Consensus in Nonlinear Opinion Dynamics

TL;DR

This work addresses how group pressure drives consensus in nonlinear, time-varying opinion dynamics with multidimensional opinions and general, opinion-dependent weights. It introduces the ODGP framework, combining an underlying averaging process with a public-opinion term and proving convergence under mild conditions: asymptotic consensus if the minimal conformity satisfies , exponential convergence when , and finite-time consensus if some . The key contributions include generalizing prior results to arbitrary underlying dynamics and public opinions within the convex hull, deriving a convergence-time bound independent of group size for positive conformity, and providing HK-specific corollaries supported by numerical simulations. The findings offer robust theoretical guarantees for consensus under diverse social-influence scenarios and have practical implications for interpreting polls and online network dynamics.

Abstract

This work extends the recent opinion dynamics model from Cheng et al., emphasizing the role of group pressure in consensus formation. We generalize the findings to incorporate social influence algorithms with general time-varying, opinion-dependent weights and multidimensional opinions, beyond bounded confidence dynamics. We demonstrate that, with uniformly positive conformity levels, group pressure consistently drives consensus and provide a tighter estimate for the convergence rate. Unlike previous models, the common public opinion in our framework can assume arbitrary forms within the convex hull of current opinions, offering flexibility applicable to real-world scenarios such as opinion polls with random participant selection. This analysis provides deeper insights into how group pressure mechanisms foster consensus under diverse conditions.
Paper Structure (12 sections, 2 theorems, 25 equations, 3 figures)

This paper contains 12 sections, 2 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Opinion Dynamics with Group Pressure
  3. Agents and their opinions
  4. Underlying Opinion Dynamics
  5. Public opinion
  6. Opinion Formation Model
  7. Consensus Caused by Group Pressure
  8. Discussion
  9. Proofs of Theorem \ref{['thm.press_converg']} and Corollary \ref{['cor.HK']}
  10. Numerical Simulations
  11. Conclusions
  12. Acknowledgements

Key Result

Theorem 1

Assume that the function $\xi^{pub}$ obeys Assumption asm.pub. Then, the following statements are valid: (i) The ODGP asymptotically converges to consensus for every initial condition if the following series diverges (ii) If, additionally, the levels of conformity remain uniformly positive $p_*(t)\geq p_0>0$ for all $t\geq 0$, the convergence to consensus is exponential, and $R(\Xi(t))\leq R_0(1-

Figures (3)

  • Figure 1: ODGP with Different $\xi^{pub}(t)$. Left part: dynamics of $\xi_1^i(t)$ (vertical axis) vs. time (horizontal axis); Right part: dynamics of $\xi_2^i(t)$ for $i \in \mathcal{V}$
  • Figure 2: ODGP with different $n$ and confidence radii $\varepsilon$, conformity level is $p=0.05$
  • Figure 3: ODGP with different $p$

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1