Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Absolute centrality in a signed Friedkin-Johnsen based model: a graphical characterisation of influence

Aashi Shrinate, Twinkle Tripathy

TL;DR

This paper studies the evolution of opinions governed by a Friedkin Johnsen (FJ) based model in arbitrary network structures with signed interactions and the absolute centrality measure is proposed to determine the overall influence of all the agents in the network.

Abstract

This paper studies the evolution of opinions governed by a Friedkin Johnsen (FJ) based model in arbitrary network structures with signed interactions. The agents contributing to the opinion formation are characterised as being influential. Initially, the agents are classified as opinion leaders and followers based on network connectivity and the nature of interactions. However, the addition of stubbornness leads to interesting behaviours wherein a non influential agent can now become influential and vice versa. Thereafter, a signal flow graph (SFG) based method is proposed to quantify the influence of an influential agents' opinions. Additionally, it helps illustrate the role played by network topology in shaping the final opinions of the agents. Based on this analysis, the absolute centrality measure is proposed to determine the overall influence of all the agents in the network. Unlike most of the existing measures, it is applicable to any network structure and considers the effect of stubbornness and antagonism. Examples are presented throughout the paper to illustrate and validate these results.

Absolute centrality in a signed Friedkin-Johnsen based model: a graphical characterisation of influence

TL;DR

This paper studies the evolution of opinions governed by a Friedkin Johnsen (FJ) based model in arbitrary network structures with signed interactions and the absolute centrality measure is proposed to determine the overall influence of all the agents in the network.

Abstract

This paper studies the evolution of opinions governed by a Friedkin Johnsen (FJ) based model in arbitrary network structures with signed interactions. The agents contributing to the opinion formation are characterised as being influential. Initially, the agents are classified as opinion leaders and followers based on network connectivity and the nature of interactions. However, the addition of stubbornness leads to interesting behaviours wherein a non influential agent can now become influential and vice versa. Thereafter, a signal flow graph (SFG) based method is proposed to quantify the influence of an influential agents' opinions. Additionally, it helps illustrate the role played by network topology in shaping the final opinions of the agents. Based on this analysis, the absolute centrality measure is proposed to determine the overall influence of all the agents in the network. Unlike most of the existing measures, it is applicable to any network structure and considers the effect of stubbornness and antagonism. Examples are presented throughout the paper to illustrate and validate these results.
Paper Structure (18 sections, 6 theorems, 26 equations, 6 figures, 1 table)

This paper contains 18 sections, 6 theorems, 26 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Notations & Preliminaries
  3. Notations
  4. Graph Preliminaries
  5. Signal Flow Graph Preliminaries
  6. Matrix Preliminaries
  7. Opinion Dynamics
  8. FJ based opinion model
  9. Classification of agents
  10. Convergence analysis
  11. Characterisation of influence using SFG
  12. Construction of SFGs
  13. Quantification of Influence
  14. Absolute Influence centrality
  15. Conclusions
  16. ...and 3 more sections

Key Result

Lemma 1

For any $M \in \mathbb{C}^{n \times n}$ and any induced norm $\|.\|$, the Gelfand's formula states that the spectral radius $\rho(M)$ of matrix $M$ is $\rho(M)=\lim_{k \to \infty}\|M^k\|^{\frac{1}{k}}$.

Figures (6)

  • Figure 1: The followers and the opinion leaders of $\mathcal{G}$ are represented by blue and orange nodes, respectively. The dotted edges denote antagonistic interactions.
  • Figure 2: A weakly connected digraph with stubborn agents $1$ and $6$.
  • Figure 3: The SFGs show the effects of each source on the rest of the nodes. Source $\mathcal{O}_4$ is excluded.
  • Figure 4: The evolution of opinions in the network given in Fig. \ref{['fig:Network_with_cycles']} governed by eqn. \ref{['eq:opinion_dynamics']}. The opinions of influential agents are indicated by solid lines and the rest by dotted lines.
  • Figure 5: Effect of sign switching in network on agent's opinions.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Lemma 1: gelfand1941normierte
  • Lemma 2
  • proof
  • Remark 1
  • Example 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Example 2
  • ...and 12 more