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A Game Approach to Multi-dimensional Opinion Dynamics in Social Networks with Stubborn Strategist Agents

Hossein B. Jond, Aykut Yıldız

TL;DR

This work finds the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure and compares the opinions evolved based on the Nash/ worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

Abstract

In a social network, individuals express their opinions on several interdependent topics, and therefore the evolution of their opinions on these topics is also mutually dependent. In this work, we propose a differential game model for the multi-dimensional opinion formation of a social network whose population of agents interacts according to a communication graph. Each individual's opinion evolves according to an aggregation of disagreements between the agent's opinions and its graph neighbors on multiple interdependent topics exposed to an unknown extraneous disturbance. For a social network with strategist agents the opinions evolve over time with respect to the minimization of a quadratic cost function that solely represents each individual's motives against the disturbance. We find the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure. Moreover, we propose a distributed implementation of the Nash/worst-case equilibrium solution. We examine the non-distributed and proposed distributed open-loop Nash/worst-case strategies on a small social network with strategist agents in a two-dimensional opinion space. Then we compare the opinions evolved based on the Nash/worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

A Game Approach to Multi-dimensional Opinion Dynamics in Social Networks with Stubborn Strategist Agents

TL;DR

This work finds the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure and compares the opinions evolved based on the Nash/ worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

Abstract

In a social network, individuals express their opinions on several interdependent topics, and therefore the evolution of their opinions on these topics is also mutually dependent. In this work, we propose a differential game model for the multi-dimensional opinion formation of a social network whose population of agents interacts according to a communication graph. Each individual's opinion evolves according to an aggregation of disagreements between the agent's opinions and its graph neighbors on multiple interdependent topics exposed to an unknown extraneous disturbance. For a social network with strategist agents the opinions evolve over time with respect to the minimization of a quadratic cost function that solely represents each individual's motives against the disturbance. We find the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure. Moreover, we propose a distributed implementation of the Nash/worst-case equilibrium solution. We examine the non-distributed and proposed distributed open-loop Nash/worst-case strategies on a small social network with strategist agents in a two-dimensional opinion space. Then we compare the opinions evolved based on the Nash/worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.
Paper Structure (11 sections, 5 theorems, 71 equations, 6 figures, 4 tables)

This paper contains 11 sections, 5 theorems, 71 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Graph theory
  4. Linear Quadratic Nash/Worst-Case Equilibria
  5. Game of Opinion Formation
  6. Nash/Worst-case Equilibrium Solution
  7. Distributed Implementation of Game Strategies
  8. Social Optimality
  9. Numerical Analysis
  10. Conclusions
  11. Commutative linear time-varying systems

Key Result

Lemma 1

Let $X^+$ and $\mathcal{R}(X)$ denote the Moore-Penrose inverse and the range, respectively, of a real matrix $X$. For nonnegative definite (or positive semidefinite) $X\in \mathit{\mathbb{R}}^{n\times n}$ and $Y\in \mathit{\mathbb{R}}^{n\times n}$, $X \overset{L}{\preceq}Y$ means $X$ is below $Y$ w

Figures (6)

  • Figure 1: Underlying communication graph topology for numerical analysis.
  • Figure 2: Opinion trajectories associated with the non-stubborn non-strategist agent model (\ref{['eq:dynamics00']}) prior to optimization. (a) A consensus of final opinions about the average opinion in the network was reached. (b) Only agents 1 and 5 reached a consensus.
  • Figure 3: Opinion trajectories associated with the game strategies and their distributed counterparts (shown with dashed lines). (a) non-stubborn agents with unrelated topics; (b) non-stubborn agents with fully coupled topics; (c) stubborn agents with unrelated topics; (d) stubborn agents with fully coupled topics.
  • Figure 4: Opinion trajectories associated with the game strategies and their distributed counterparts with disturbances $r_{\varpi_i}=0.5$ for $i=1,\cdots,5$.
  • Figure 5: Opinion trajectories associated with socially optimal actions (shown with a dash-dotted line) versus opinion trajectories associated with game strategies.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof