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On the Analysis of the DeGroot-Friedkin Model with Dynamic Relative Interaction Matrices

Mengbin Ye, Ji Liu, Brian David Outram Anderson, Changbin Yu, Tamer Başar

TL;DR

This paper analyses the DeGroot-Friedkin model for evolution of the individuals’ social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices), and shows that individuals‘ social powers admit a periodic solution.

Abstract

This paper analyses the DeGroot-Friedkin model for evolution of the individuals' social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices). The DeGroot-Friedkin model describes how individual social power (self-appraisal, self-weight) evolves as a network of individuals discuss a sequence of issues. We seek to study dynamically changing relative interactions because interactions may change depending on the issue being discussed. In order to explore the problem in detail, two different cases of issue-dependent network topologies are studied. First, if the topology varies between issues in a periodic manner, it is shown that the individuals' self-appraisals admit a periodic solution. Second, if the topology changes arbitrarily, under the assumption that each relative interaction matrix is doubly stochastic and irreducible, the individuals' self-appraisals asymptotically converge to a unique non-trivial equilibrium.

On the Analysis of the DeGroot-Friedkin Model with Dynamic Relative Interaction Matrices

TL;DR

This paper analyses the DeGroot-Friedkin model for evolution of the individuals’ social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices), and shows that individuals‘ social powers admit a periodic solution.

Abstract

This paper analyses the DeGroot-Friedkin model for evolution of the individuals' social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices). The DeGroot-Friedkin model describes how individual social power (self-appraisal, self-weight) evolves as a network of individuals discuss a sequence of issues. We seek to study dynamically changing relative interactions because interactions may change depending on the issue being discussed. In order to explore the problem in detail, two different cases of issue-dependent network topologies are studied. First, if the topology varies between issues in a periodic manner, it is shown that the individuals' self-appraisals admit a periodic solution. Second, if the topology changes arbitrarily, under the assumption that each relative interaction matrix is doubly stochastic and irreducible, the individuals' self-appraisals asymptotically converge to a unique non-trivial equilibrium.

Paper Structure

This paper contains 16 sections, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and Problem Statement
  3. Graph Theory
  4. Modelling the Dynamics of the Social Network
  5. DeGroot Consensus of Opinions
  6. Friedkin's Self-Appraisal Model for Determining Self-Weight
  7. Problem Formulation
  8. Periodic Switching
  9. Transformation into a Time-Invariant System
  10. Existence of a Periodic Sequence
  11. Generalisation to $M$ Topologies
  12. Arbitrary Switching
  13. Simulations
  14. Conclusion
  15. Proof of Lemma \ref{['lem:properties_F_bar']}
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Evolution of an individual's self-weight for $\boldsymbol{C}_1$ and $\boldsymbol{C}_2$.
  • Figure 2: Evolution of an individual's self-weight for $\boldsymbol{C}_1$ and $\boldsymbol{C}_2$, different initial conditions.
  • Figure 3: Evolution of an individual's self-weight for $\boldsymbol{C}_1$, $\boldsymbol{C}_2$ and $\boldsymbol{C}_3$.
  • Figure 4: Evolution of an individual's self-weight when $\boldsymbol{C}(s)$ varies arbitrarily between $\boldsymbol{C}_4$ and $\boldsymbol{C}_5$.