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Evolution of Social Power in Social Networks with Dynamic Topology

Mengbin Ye, Ji Liu, Brian D. O. Anderson, Changbin Yu, Tamer Başar

TL;DR

This paper shows that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence, and provides an explicit upper bound on an individuals' social power as the number of issues discussed tends to infinity.

Abstract

The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results.

Evolution of Social Power in Social Networks with Dynamic Topology

TL;DR

This paper shows that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence, and provides an explicit upper bound on an individuals' social power as the number of issues discussed tends to infinity.

Abstract

The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results.

Paper Structure

This paper contains 21 sections, 17 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions of This Paper
  3. Structure of the Rest of the Paper
  4. Background and Problem Statement
  5. Graph Theory
  6. The DeGroot-Friedkin Model
  7. DeGroot Consensus of Opinions
  8. Friedkin's Self-Appraisal Model for Determining Self-Weight
  9. Quantitative Aspects of the Dynamic Topology Problem
  10. Contraction Analysis for Nonlinear Systems
  11. Contraction Analysis for Constant $\boldsymbol{C}$
  12. Fundamental Contraction Analysis
  13. Extending the Contraction-like Analysis
  14. Convergence Rate for a Set of $\boldsymbol{C}$ Matrices
  15. Dynamic Relative Interaction Topology
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{n\times n}$ be symmetric. If $\boldsymbol{A}$ is positive definite, then $\boldsymbol{AB}$ is diagonalizable and has real eigenvalues. If, in addition, $\boldsymbol{B}$ is positive definite or positive semidefinite, then the eigenvalues of $\boldsy

Figures (3)

  • Figure 1: Evolution of individuals' social powers $\boldsymbol{x}(s)$ for initial condition set $\widehat{\boldsymbol{x}}(0)$.
  • Figure 2: Evolution of individuals' social powers $\boldsymbol{x}(s)$ for initial condition set $\widetilde{\boldsymbol{x}}(0)$.
  • Figure 3: Evolution of selected individuals' social powers $x_i(s)$: a comparison of different initial condition sets $\widehat{\boldsymbol{x}}(0)$ and $\widetilde{\boldsymbol{x}}(0)$.

Theorems & Definitions (43)

  • Lemma 1: Corollary 7.6.2 in horn2012matrixbook
  • Remark 1: Time-scales
  • Remark 2: Social Power
  • Definition 1: Star topology
  • Lemma 2: Lemma 3.2 in jia2015opinion_SIAM
  • Theorem 1: Theorem 4.1 in jia2015opinion_SIAM
  • Definition 2: Generalised Contraction Region
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 33 more