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On a Co-evolving Opinion-Leadership Model in Social Networks

Martina Alutto, Lorenzo Zino, Karl H. Johansson, Angela Fontan

Abstract

Leadership in social groups is often a dynamic characteristic that emerges from interactions and opinion exchange. Empirical evidence suggests that individuals with strong opinions tend to gain influence, at the same time maintaining alignment with the social context is crucial for sustained leadership. Motivated by the social psychology literature that supports these empirical observations, we propose a novel dynamical system in which opinions and leadership co-evolve within a social network. Our model extends the Friedkin-Johnsen framework by making susceptibility to peer influence time-dependent, turning it into the leadership variable. Leadership strengthens when an agent holds strong yet socially aligned opinions, and declines when such alignment is lost, capturing the trade-off between conviction and social acceptance. After illustrating the emergent behavior of this complex system, we formally analyze the coupled dynamics, establishing sufficient conditions for convergence to a non-trivial equilibrium, and examining two time-scale separation regimes reflecting scenarios where opinion and leadership evolve at different speeds.

On a Co-evolving Opinion-Leadership Model in Social Networks

Abstract

Leadership in social groups is often a dynamic characteristic that emerges from interactions and opinion exchange. Empirical evidence suggests that individuals with strong opinions tend to gain influence, at the same time maintaining alignment with the social context is crucial for sustained leadership. Motivated by the social psychology literature that supports these empirical observations, we propose a novel dynamical system in which opinions and leadership co-evolve within a social network. Our model extends the Friedkin-Johnsen framework by making susceptibility to peer influence time-dependent, turning it into the leadership variable. Leadership strengthens when an agent holds strong yet socially aligned opinions, and declines when such alignment is lost, capturing the trade-off between conviction and social acceptance. After illustrating the emergent behavior of this complex system, we formally analyze the coupled dynamics, establishing sufficient conditions for convergence to a non-trivial equilibrium, and examining two time-scale separation regimes reflecting scenarios where opinion and leadership evolve at different speeds.

Paper Structure

This paper contains 11 sections, 32 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model Description
  3. Notation
  4. Illustrative Examples
  5. Three-node network
  6. Impact of model parameters
  7. Equilibria and Stability Analysis
  8. Time-scale Separation
  9. Fast-leadership regime
  10. Slow-leadership regime
  11. Conclusion

Figures (6)

  • Figure 1: Numerical simulation of the three-agent opinion-leadership model with $\alpha = 0.6$ and $\beta =1$. (a) Symmetric case: agent $2$ remains neutral while agents $1$ and $3$ converge to moderated polarized opinions. (b)–(d) Perturbed scenarios showing how asymmetry in the opinion initialization or interaction strengths breaks polarization symmetry and modifies leadership emergence.
  • Figure 2: Simulation of the opinion-leadership model \ref{['eq:leader-model']} with $8$ agents with $\alpha = 0.6$ and (a,b) $\beta = 0.1$ or (c,d) $\beta = 1$.
  • Figure 3: Region of admissible parameter values $(\varepsilon, \rho)$ for which condition \ref{['eq:cond-norm-infty']} holds true.
  • Figure 4: Simulation of the model in \ref{['eq:leader-model-timescale']} under the fast-leadership regime ($\tau \to 0$).
  • Figure 5: Simulation of the model in \ref{['eq:leader-model-timescale']} under the slow-leadership regime ($\tau \to \infty$).
  • ...and 1 more figures