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Dynamical models for distributed social power perception in Friedkin-Johnsen influence networks

Ye Tian, Angela Fontan, Yu Kawano, Wei Zhang, Kenji Kashima, Karl H. Johansson

Abstract

Social power quantifies the ability of individuals to influence others and plays a central role in social influence networks. Yet, computing social power typically requires global knowledge and significant computational or storage capability, especially in large-scale networks with stubborn individuals. In this paper, we propose a distributed perception mechanism based on the Friedkin-Johnsen opinion dynamics that enables individuals to estimate their true social power through local interactions. The mechanism starts from independent initial perceptions and relies only on local information: each individual only needs to know its neighbors' stubbornness and the influence weights they accord. We provide rigorous dynamical system analysis that characterizes equilibria, invariant sets, and convergence. Conditions are established for convergence to the true social power in both the static setting with fixed influence weights and the reflected-appraisal setting where influence weights coevolve with perceptions. The proposed mechanism remains reliable under extreme initial perceptions, disconnected influence networks, reflected-appraisal coupling, and variations in timescales. Numerical examples illustrate our results.

Dynamical models for distributed social power perception in Friedkin-Johnsen influence networks

Abstract

Social power quantifies the ability of individuals to influence others and plays a central role in social influence networks. Yet, computing social power typically requires global knowledge and significant computational or storage capability, especially in large-scale networks with stubborn individuals. In this paper, we propose a distributed perception mechanism based on the Friedkin-Johnsen opinion dynamics that enables individuals to estimate their true social power through local interactions. The mechanism starts from independent initial perceptions and relies only on local information: each individual only needs to know its neighbors' stubbornness and the influence weights they accord. We provide rigorous dynamical system analysis that characterizes equilibria, invariant sets, and convergence. Conditions are established for convergence to the true social power in both the static setting with fixed influence weights and the reflected-appraisal setting where influence weights coevolve with perceptions. The proposed mechanism remains reliable under extreme initial perceptions, disconnected influence networks, reflected-appraisal coupling, and variations in timescales. Numerical examples illustrate our results.

Paper Structure

This paper contains 23 sections, 15 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Friedkin-Johnsen opinion dynamics
  4. True social power and its evolution
  5. Perceived social power and distributed perception
  6. Distributed perception of static social power
  7. Distributed perception of social power with reflected appraisals
  8. Distributed perceived-reflected-appraisal dynamics
  9. Equilibria: existence, uniqueness and dominance structure
  10. Invariant sets and convergence
  11. Convergence in structured networks: Star topology and homogeneous stubbornness
  12. Star topology
  13. Homogeneous stubbornness and reflected-appraisal PageRank
  14. Conclusion
  15. Auxiliary lemmas
  16. ...and 8 more sections

Key Result

Proposition 1

(Convergence of perception dynamics) Suppose that Assumption A1 holds. Then all trajectories of system e6 starting from $p(0)\in\mathbb{R}^{n}$ converge exponentially to the true social power $x$ of the FJ model given by e4. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Dynamics of true and perceived social power under reflected appraisal. The shaded region marks the information required to compute $x_i$ (true social power) or to update $p_{i}$ (perceived social power) in each layer.
  • Figure 2: Trajectories of system \ref{['e6']} under various initializations.
  • Figure 3: (a) Feasible region of $(a_2,a_3)$ satisfying \ref{['equ:15']} and \ref{['equ:16']} for $n=3$, $a_1=0$, and the given matrix $C$. (b) Trajectories of system \ref{['e11']} from $p(0)$ satisfying \ref{['Mini']} and system \ref{['e5']} from $x(0)\in\Delta_n$.
  • Figure 4: Trajectories of system \ref{['e11']} under: (a) star topology with fully stubborn center node; (b) star topology with partially stubborn center node; (c) homogeneous stubbornness.

Theorems & Definitions (25)

  • Remark 1
  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Theorem 1
  • Proposition 3
  • Theorem 2
  • Example 2
  • ...and 15 more