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On Online Control of Opinion Dynamics

Sheryl Paul, Leslie Cruz Juarez, Jyotirmoy V. Deshmukh, Ketan Savla

Abstract

Networked multi-agent dynamical systems have been used to model how individual opinions evolve over time due to the opinions of other agents in the network. Particularly, such a model has been used to study how a planning agent can be used to steer opinions in a desired direction through repeated, budgeted interventions. In this paper, we consider the problem where individuals' susceptibilities to external influences are unknown. We propose an online algorithm that alternates between estimating this susceptibility parameter, and using the current estimate to drive the opinion to a desired target. We provide conditions that guarantee stability and convergence to the desired target opinion when the planning agent faces budgetary or temporal constraints. Our analysis shows that the key advantage of estimating the susceptibility parameter is that it helps achieve near-optimal convergence to the target opinion given a finite amount of intervention rounds, and, for a given intervention budget, quantifies how close the opinion can get to the desired target.

On Online Control of Opinion Dynamics

Abstract

Networked multi-agent dynamical systems have been used to model how individual opinions evolve over time due to the opinions of other agents in the network. Particularly, such a model has been used to study how a planning agent can be used to steer opinions in a desired direction through repeated, budgeted interventions. In this paper, we consider the problem where individuals' susceptibilities to external influences are unknown. We propose an online algorithm that alternates between estimating this susceptibility parameter, and using the current estimate to drive the opinion to a desired target. We provide conditions that guarantee stability and convergence to the desired target opinion when the planning agent faces budgetary or temporal constraints. Our analysis shows that the key advantage of estimating the susceptibility parameter is that it helps achieve near-optimal convergence to the target opinion given a finite amount of intervention rounds, and, for a given intervention budget, quantifies how close the opinion can get to the desired target.
Paper Structure (9 sections, 4 theorems, 32 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 4 theorems, 32 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. ANALYSIS OF SYSTEM DYNAMICS
  4. Stability Analysis under Bounded Control Input
  5. Feasibility Analysis of Control Cost and Accuracy
  6. PARAMETER ESTIMATION
  7. ONLINE CONTROL ALGORITHM
  8. SIMULATIONS
  9. CONCLUSIONS

Key Result

Theorem 1

If $V \in \mathbb{R}^{n \times n}$ is row-stochastic matrix, i.e., $V\mathbf{1}=\mathbf{1}$ and $v_{ij} \geq 0, \ \forall i,j$, and the control vector $u(t)$ satisfies $0\le h_i u_i(t) \le 1 \ \forall i$, for $\forall t$, assuming that the opinion vector $x(t-1) \text{ lies in the subset } [0,1]^n

Figures (3)

  • Figure 1: Final error $\epsilon(T)$ vs. control cost $c_u(T)$ under known $\theta$ vs. adaptive control using $\hat{\theta}(t)$.
  • Figure 2: Final error $\epsilon$ across costs comparing the IODSFC baseline and our method.
  • Figure 3: Comparison of our method with the NRS-OFO method under different experimental setups. (a) Step-wise Gradient Descent. (b) Steady State Errors under fixed budgets. (c) Parameter estimation of $\theta$ using \ref{['eq:param-est-algo']} with gradient descent versus with our proposed method.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2: Stability of the desired equilibrium for admissible initial conditions
  • proof
  • Remark 3: Joint Spectral Radius Argument.
  • Remark 4: System behavior without control input.
  • Lemma 1: Convergence of Parameter Estimation Error
  • proof
  • Proposition 1: Convergence of combined error
  • proof
  • ...and 1 more