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Containment Control Approach for Steering Opinion in a Social Network

Hossein Rastgoftar

TL;DR

The paper formulates multidimensional opinion dynamics under the Friedkin–Johnsen model as a containment-control problem with stubborn agents acting as leaders who specify a desired opinion distribution via a distributed reward. It develops a decentralized scheme for regular agents to acquire biases and influence weights from local rewards, and proves convergence under both irreducible and reducible network topologies, including time-varying and random communications. The authors demonstrate finite-time convergence and containment steering through simulations in DNN-based reducible networks and irreducible networks, illustrating the practical potential for decentralized opinion steering. This work provides a theoretical and computational framework for steering social opinions with provable stability and convergence, enabling robust, leader-guided influence in complex networks.

Abstract

The paper studies the problem of steering multi-dimensional opinion in a social network. Assuming the society of desire consists of stubborn and regular agents, stubborn agents are considered as leaders who specify the desired opinion distribution as a distributed reward or utility function. In this context, each regular agent is seen as a follower, updating its bias on the initial opinion and influence weights by averaging their observations of the rewards their influencers have received. Assuming random graphs with reducible and irreducible topology specify the influences on regular agents, opinion evolution is represented as a containment control problem in which stability and convergence to the final opinion are proven.

Containment Control Approach for Steering Opinion in a Social Network

TL;DR

The paper formulates multidimensional opinion dynamics under the Friedkin–Johnsen model as a containment-control problem with stubborn agents acting as leaders who specify a desired opinion distribution via a distributed reward. It develops a decentralized scheme for regular agents to acquire biases and influence weights from local rewards, and proves convergence under both irreducible and reducible network topologies, including time-varying and random communications. The authors demonstrate finite-time convergence and containment steering through simulations in DNN-based reducible networks and irreducible networks, illustrating the practical potential for decentralized opinion steering. This work provides a theoretical and computational framework for steering social opinions with provable stability and convergence, enabling robust, leader-guided influence in complex networks.

Abstract

The paper studies the problem of steering multi-dimensional opinion in a social network. Assuming the society of desire consists of stubborn and regular agents, stubborn agents are considered as leaders who specify the desired opinion distribution as a distributed reward or utility function. In this context, each regular agent is seen as a follower, updating its bias on the initial opinion and influence weights by averaging their observations of the rewards their influencers have received. Assuming random graphs with reducible and irreducible topology specify the influences on regular agents, opinion evolution is represented as a containment control problem in which stability and convergence to the final opinion are proven.

Paper Structure

This paper contains 15 sections, 5 theorems, 50 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Problem Statement
  5. Opinion Evolution Dynamics
  6. Opinion Evolution under Irreducible Network
  7. Opinion Evolution under Reducible Network
  8. Decentralized Acquisition of Biases and Influences
  9. Simulation Results
  10. Opinion Evolution under DNN-Based Reducible Network
  11. Opinion Evolution under Irreducible Network
  12. Convergence and Opinion Steering under Irreducible Network
  13. Opinion Evolution under Time-Varying and Random Network
  14. Conclusion
  15. Acknowledgement

Key Result

Theorem 1

If graph $\mathcal{G}$ is defined such that there exists at least one path from agent, then, $i\in \mathcal{V}_R$, then, dynamics networkopiniondynamics is stable.

Figures (12)

  • Figure 1: The example inte-agent influences shown in sub-figure (a) is converted into a DNN shown in sub-figure (b).
  • Figure 2: Initial distribution of of $\mathcal{V}$ in the opinion space $o_1-o_2$. The red and black nodes represent "stubborn" and "regular" agents, respectively.
  • Figure 3: Inter-agent communication is defined by a deep neural network with $M=3$ internal layers.
  • Figure 4: The reward distribution over the opinion space and the final opinion of the community.
  • Figure 5: The $o_1$ (first) component of opinion of all regular agents versus discrete time $k$.It is seen that the first component of all regular agents converges to its final value in $M=3$ time steps.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 9 more