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Coevolutionary Dynamics of Actions and Opinions in Social Networks

Hassan Dehghani Aghbolagh, Mengbin Ye, Lorenzo Zino, Ming Cao, Zhiyong Chen

TL;DR

This article establishes global convergence to the Nash equilibria of the game, proving that actions converge in a finite number of time steps, while opinions converge asymptotically, and proves that the coevolutionary dynamics is an ordinal potential game, enabling analysis via potential game theory.

Abstract

Empirical studies suggest a deep intertwining between opinion formation and decision-making processes, but these have been treated as separate problems in the study of dynamical models for social networks. In this paper, we bridge the gap in the literature by proposing a novel coevolutionary model, in which each individual selects an action from a binary set and has an opinion on which action they prefer. Actions and opinions coevolve on a two-layer network. For homogeneous parameters, undirected networks, and under reasonable assumptions on the asynchronous updating mechanics, we prove that the coevolutionary dynamics is an ordinal potential game, enabling analysis via potential game theory. Specifically, we establish global convergence to the Nash equilibria of the game, proving that actions converge in a finite number of time steps, while opinions converge asymptotically. Next, we provide sufficient conditions for the existence of, and convergence to, polarized equilibria, whereby the population splits into two communities, each selecting and supporting one of the actions. Finally, we use simulations to examine the social psychological phenomenon of pluralistic ignorance.

Coevolutionary Dynamics of Actions and Opinions in Social Networks

TL;DR

This article establishes global convergence to the Nash equilibria of the game, proving that actions converge in a finite number of time steps, while opinions converge asymptotically, and proves that the coevolutionary dynamics is an ordinal potential game, enabling analysis via potential game theory.

Abstract

Empirical studies suggest a deep intertwining between opinion formation and decision-making processes, but these have been treated as separate problems in the study of dynamical models for social networks. In this paper, we bridge the gap in the literature by proposing a novel coevolutionary model, in which each individual selects an action from a binary set and has an opinion on which action they prefer. Actions and opinions coevolve on a two-layer network. For homogeneous parameters, undirected networks, and under reasonable assumptions on the asynchronous updating mechanics, we prove that the coevolutionary dynamics is an ordinal potential game, enabling analysis via potential game theory. Specifically, we establish global convergence to the Nash equilibria of the game, proving that actions converge in a finite number of time steps, while opinions converge asymptotically. Next, we provide sufficient conditions for the existence of, and convergence to, polarized equilibria, whereby the population splits into two communities, each selecting and supporting one of the actions. Finally, we use simulations to examine the social psychological phenomenon of pluralistic ignorance.
Paper Structure (24 sections, 8 theorems, 64 equations, 4 figures, 1 table)

This paper contains 24 sections, 8 theorems, 64 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Graph Theory
  4. Game Theory
  5. Decision-Making and Opinion Dynamics
  6. Decision-Making as a Network Coordination Game
  7. Friedkin-Johnsen Opinion Dynamics Model
  8. Motivation for a Coevolutionary Framework
  9. Coevolutionary Model
  10. Setting
  11. Payoff Function
  12. Dynamics
  13. Convergence Analysis
  14. Emergent Real-world Phenomena
  15. Polarization
  16. ...and 9 more sections

Key Result

Proposition 1

For the coevolutionary game $\gamma=(\mathcal{V},\mathcal{A},\boldsymbol{f})$ on the two-layer network $\mathcal{G}$, with the $i$th entry of $\boldsymbol{f}$ defined in Eq. (eq:f_function), and the function $\delta_i(\boldsymbol{z})$ defined in Eq. (eq:delta), consider a player $i\in\mathcal{V}$. S If $\delta_i(\boldsymbol{z})=0$, then $\mathcal{B}_i(f_i(\cdot,\boldsymbol{z}(t)))=\{(+1,\zeta_o),(

Figures (4)

  • Figure 1: The two-layer network structure utilized in the model.
  • Figure 2: Schematic of the coevolutionary model.
  • Figure 3: Network and simulation of Example \ref{['ex:polarization']}. In (a) and (b), the network and the actions are shown at time $t=0$ and $t=150$, respectively. Nodes are colored in green if $\text{sgn}(y_i)=x_i=+1$, in red if $\text{sgn}(y_i)=x_i=-1$, and in yellow if $\text{sgn}(y_i)\neq x_i$. In (c), the temporal evolution of the opinion are shown to converge to a polarized scenario.
  • Figure 4: Simulation illustrating the emergence and resolution of pluralistic ignorance among female and male agents, respectively, closely following the findings in prentice1993pluralistic.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Remark 2
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Theorem 1
  • Definition 3
  • ...and 5 more