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Social power evolution in influence networks with stubborn individuals

Ye Tian, Peng Jia, Anahita Mirtabatabaei, Long Wang, Noah E. Friedkin, Francesco Bullo

TL;DR

This work extends the DeGroot-Friedkin model by incorporating Friedkin-Johnsen opinion dynamics to study social power evolution in networks with stubborn individuals. It presents two formulations—over issue sequences and on a single issue—showing they share the same equilibrium social power $x^*$ and deriving conditions for existence, positivity, and uniqueness under general and star topologies. The analysis yields contraction-based existence and exponential convergence results, and demonstrates that autocratic outcomes cannot arise while democratic equilibria can be achieved across network structures; simulations corroborate the theory and motivate a conjecture on global attractivity. Overall, the paper reveals that stubbornness anchors social power via reflected appraisal, with robust democratic outcomes and broad applicability to influence dynamics in complex networks.

Abstract

This paper studies the evolution of social power in influence networks with stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the reflected appraisal mechanism, two models are proposed over issue sequences and over a single issue, respectively. These models generalize the original DeGroot-Friedkin (DF) model by including stubbornness. To the best of our knowledge, this paper is the first attempt to investigate the social power evolution of stubborn individuals basing on the reflected appraisal mechanism. Properties of equilibria and convergence are provided. We show that the models have same equilibrium social power and convergence property, where the equilibrium social power depends only upon interpersonal influence and individuals' stubbornness. Roughly speaking, more stubborn individual has more equilibrium social power. Moreover, unlike the DF model without stubbornness, we prove that for the models with stubbornness, autocracy can never be achieved, while democracy can be achieved under any network topology.

Social power evolution in influence networks with stubborn individuals

TL;DR

This work extends the DeGroot-Friedkin model by incorporating Friedkin-Johnsen opinion dynamics to study social power evolution in networks with stubborn individuals. It presents two formulations—over issue sequences and on a single issue—showing they share the same equilibrium social power and deriving conditions for existence, positivity, and uniqueness under general and star topologies. The analysis yields contraction-based existence and exponential convergence results, and demonstrates that autocratic outcomes cannot arise while democratic equilibria can be achieved across network structures; simulations corroborate the theory and motivate a conjecture on global attractivity. Overall, the paper reveals that stubbornness anchors social power via reflected appraisal, with robust democratic outcomes and broad applicability to influence dynamics in complex networks.

Abstract

This paper studies the evolution of social power in influence networks with stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the reflected appraisal mechanism, two models are proposed over issue sequences and over a single issue, respectively. These models generalize the original DeGroot-Friedkin (DF) model by including stubbornness. To the best of our knowledge, this paper is the first attempt to investigate the social power evolution of stubborn individuals basing on the reflected appraisal mechanism. Properties of equilibria and convergence are provided. We show that the models have same equilibrium social power and convergence property, where the equilibrium social power depends only upon interpersonal influence and individuals' stubbornness. Roughly speaking, more stubborn individual has more equilibrium social power. Moreover, unlike the DF model without stubbornness, we prove that for the models with stubbornness, autocracy can never be achieved, while democracy can be achieved under any network topology.

Paper Structure

This paper contains 22 sections, 13 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Modeling
  3. The DF model with stubborn individuals over issue sequences
  4. The DF model with stubborn individuals over a single issue
  5. Equilibrium analysis
  6. Equivalence of equilibrium social power
  7. Properties of equilibrium social power with general topology
  8. Properties of equilibrium social power with star topology
  9. Convergence analysis
  10. Convergence of the DF model with stubborn individuals over issue sequences
  11. Convergence of the DF model with stubborn individuals over a single issue
  12. Simulations and conjecture
  13. Conclusions
  14. Proof of Lemma \ref{['L4']}
  15. Proof of Lemma \ref{['L4.1']}
  16. ...and 7 more sections

Key Result

Lemma 1

(Equivalence of equilibrium social power) Suppose that Assumption A1 holds, system e4 and e14 have the same relative interaction matrix $C$ and susceptibility matrix $\Theta$. Then, $x^{*}$ is an equilibrium of system e4 if and only if for $V^{*}=(I_{n}-\Theta W(x^{*}))^{-1}(I_{n}-\Theta)\in\Gamma_{

Figures (2)

  • Figure 1: $50$ runs of trajectories of $x(s)$ under different $\Theta$.
  • Figure 2: The trajectories for $6$ nodes of systems \ref{['e4']} and \ref{['e14']} beginning at $100$ samples of initial social power with $3$ samples of matrix pairs $(C^{i},\Theta^{i})$. The solid lines and dot lines dipict the trajectories of system \ref{['e4']} and \ref{['e14']}, respectively.

Theorems & Definitions (19)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 3
  • Corollary 1
  • Corollary 2
  • ...and 9 more