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Opinion Dynamics in Social Multiplex Networks with Mono and Bi-directional Interactions in the Presence of Leaders

Amirreza Talebi, Sayed Pedram Haeri Boroujeni, Abolfazl Razi

TL;DR

The paper addresses how opinions evolve on a two-layer multiplex network under coordination-game dynamics, considering mono- and bi-directional interactions and the presence of leaders. It models a layer-switching regime with one layer always active and the other active on even steps, and proves convergence to either the leader's initial opinion or a convex combination of agents' opinions, depending on the existence of a leader. The analysis combines graph-theoretic conditions (e.g., directed spanning trees) with Markov-chain decompositions to derive convergence results and rate bounds that depend on transient submatrices. The results show that designated leaders accelerate convergence and that mono-directional interactions yield faster consensus than bidirectional ones, with implications for controlling opinion dynamics in dynamic multiplex networks.

Abstract

We delve into the dynamics of opinions within a multiplex network using coordination games, where agents communicate either in a one-way or two-way interactions, and where a designated leader may be present. By employing graph theory and Markov chains, we illustrate that despite non-positive diagonal elements in transition probability matrices or decomposable layers, opinions generally converge under specific conditions, leading to a consensus. We further scrutinize the convergence rates of opinion dynamics in networks with one-way versus two-way interactions. We find that in networks with a designated leader, opinions converge towards the initial opinion of the leader, whereas in networks without a designated leader, opinions converge to a convex combination of the opinions of agents. Moreover, we emphasize the crucial role of designated leaders in steering opinion convergence within the network. Our experimental findings corroborate that the presence of leaders expedites convergence, with mono-directional interactions exhibiting notably faster convergence rates compared to bidirectional interactions.

Opinion Dynamics in Social Multiplex Networks with Mono and Bi-directional Interactions in the Presence of Leaders

TL;DR

The paper addresses how opinions evolve on a two-layer multiplex network under coordination-game dynamics, considering mono- and bi-directional interactions and the presence of leaders. It models a layer-switching regime with one layer always active and the other active on even steps, and proves convergence to either the leader's initial opinion or a convex combination of agents' opinions, depending on the existence of a leader. The analysis combines graph-theoretic conditions (e.g., directed spanning trees) with Markov-chain decompositions to derive convergence results and rate bounds that depend on transient submatrices. The results show that designated leaders accelerate convergence and that mono-directional interactions yield faster consensus than bidirectional ones, with implications for controlling opinion dynamics in dynamic multiplex networks.

Abstract

We delve into the dynamics of opinions within a multiplex network using coordination games, where agents communicate either in a one-way or two-way interactions, and where a designated leader may be present. By employing graph theory and Markov chains, we illustrate that despite non-positive diagonal elements in transition probability matrices or decomposable layers, opinions generally converge under specific conditions, leading to a consensus. We further scrutinize the convergence rates of opinion dynamics in networks with one-way versus two-way interactions. We find that in networks with a designated leader, opinions converge towards the initial opinion of the leader, whereas in networks without a designated leader, opinions converge to a convex combination of the opinions of agents. Moreover, we emphasize the crucial role of designated leaders in steering opinion convergence within the network. Our experimental findings corroborate that the presence of leaders expedites convergence, with mono-directional interactions exhibiting notably faster convergence rates compared to bidirectional interactions.
Paper Structure (13 sections, 2 theorems, 13 equations, 7 figures)

This paper contains 13 sections, 2 theorems, 13 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Related Work
  4. Model and Preliminaries
  5. The multiplex network
  6. The adjacency matrices
  7. Agents' opinion dynamics
  8. Convergence of the Opinion Profile
  9. Convergence Rate
  10. Numerical Results
  11. Conclusions
  12. Declaration of competing interest
  13. Data and Code availability

Key Result

Lemma 1

The convergence rate of the opinion dynamics, characterized by the weighted adjacency matrix $C$, can be described as follows: Here, $\mathbf{\bar{x}}$ is the final converged opinion profile, $U>0$ is a constant, and $q$ represents the joint spectral radius of the multiplication of adjacency matrices of the layers at different time steps.

Figures (7)

  • Figure 1: An example of a two-layer multiplex network: Agent $A$ assumes a leadership role on Layer 1 but not on Layer 2. The neighbors of agent $B$ include agents $A$ and $D$ on both layers.
  • Figure 2: An example of a two-layer multiplex network at $t=2$ when there is a leader in the union of layers.
  • Figure 3: An example of a two-layer multiplex network at $t=2$ without any leader in the network.
  • Figure 4: Convergence rate for example 1 network topology with mono-directional interactions
  • Figure 5: Convergence rate for example 2 network topology with mono-directional interactions
  • ...and 2 more figures

Theorems & Definitions (4)

  • proof
  • Lemma 1: blondel2005convergence
  • Proposition 1
  • proof