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Model Predictive Control for Coupled Adoption-Opinion Dynamics

Martina Alutto, Qiulin Xu, Fabrizio Dabbene, Hideaki Ishii, Chiara Ravazzi

TL;DR

This work develops a coupled adoption–opinion model on a multilayer network, combining a three-state SAD adoption framework on a physical network with Friedkin–Johnsen-inspired opinion dynamics on a social layer. Adoption and opinion influence each other through inter-layer feedback, and the model is analyzed for adoption-free and adoption-diffused equilibria using a reproduction-number $R_0^A(x)$ that depends on opinions. The paper then introduces a nonlinear Model Predictive Control (MPC) scheme that shapes opinions via a bounded control $u(t)$ to indirectly maximize adoption, with a finite-horizon optimization and a receding-horizon implementation. Numerical simulations demonstrate that, in the absence of control, adoption stagnates, whereas MPC-based interventions sustain and enhance adoption across communities, outperforming a constant policy in effectiveness per unit effort.

Abstract

This paper investigates an optimal control problem for an adoption-opinion model that couples opinion dynamics with a compartmental adoption framework on a multilayer network to study the diffusion of sustainable behaviors. Adoption evolves through social contagion and perceived benefits, while opinions are shaped by social interactions and feedback from adoption levels. Individuals may also stop adopting virtuous behavior due to external constraints or shifting perceptions, affecting overall diffusion. After the stability analysis of equilibria, both in the presence and absence of adopters, we introduce a Model Predictive Control (MPC) framework that optimizes interventions by shaping opinions rather than directly enforcing adoption. This nudge-based control strategy allows policymakers to influence diffusion indirectly, making interventions more effective and scalable. Numerical simulations demonstrate that, in the absence of control, adoption stagnates, whereas MPC-driven interventions sustain and enhance adoption across communities.

Model Predictive Control for Coupled Adoption-Opinion Dynamics

TL;DR

This work develops a coupled adoption–opinion model on a multilayer network, combining a three-state SAD adoption framework on a physical network with Friedkin–Johnsen-inspired opinion dynamics on a social layer. Adoption and opinion influence each other through inter-layer feedback, and the model is analyzed for adoption-free and adoption-diffused equilibria using a reproduction-number that depends on opinions. The paper then introduces a nonlinear Model Predictive Control (MPC) scheme that shapes opinions via a bounded control to indirectly maximize adoption, with a finite-horizon optimization and a receding-horizon implementation. Numerical simulations demonstrate that, in the absence of control, adoption stagnates, whereas MPC-based interventions sustain and enhance adoption across communities, outperforming a constant policy in effectiveness per unit effort.

Abstract

This paper investigates an optimal control problem for an adoption-opinion model that couples opinion dynamics with a compartmental adoption framework on a multilayer network to study the diffusion of sustainable behaviors. Adoption evolves through social contagion and perceived benefits, while opinions are shaped by social interactions and feedback from adoption levels. Individuals may also stop adopting virtuous behavior due to external constraints or shifting perceptions, affecting overall diffusion. After the stability analysis of equilibria, both in the presence and absence of adopters, we introduce a Model Predictive Control (MPC) framework that optimizes interventions by shaping opinions rather than directly enforcing adoption. This nudge-based control strategy allows policymakers to influence diffusion indirectly, making interventions more effective and scalable. Numerical simulations demonstrate that, in the absence of control, adoption stagnates, whereas MPC-driven interventions sustain and enhance adoption across communities.
Paper Structure (12 sections, 5 theorems, 11 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 5 theorems, 11 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Model Description
  4. Adoption dynamics
  5. Opinion dynamics
  6. Coupled adoption-opinion model
  7. Stability results
  8. Adoption-free equilibrium
  9. Adoption-diffused equilibrium
  10. Model Predictive control
  11. Simulation Results
  12. Conclusion

Key Result

Proposition 1

Consider the adoption-opinion model eq:adoption-model-eq:opinion-model under Assumption ass:ass1. Then, if $s(0), a(0), d(0)$ in $[0,1]^{\mathcal{V}}$ and $s(0)+ a(0)+ d(0) = \mathds{1}$, then $s(t), a(t),d(t)$ in $[0,1]^{\mathcal{V}}$ and $s(t)+a(t)+d(t)=\mathds{1}$ for all $t\geq0$. Moreover, if $

Figures (4)

  • Figure 1: (a) The bilayer network of the coupled adoption-opinion model. (b) Adoption model with three states and various transition parameters.
  • Figure 2: Numerical simulation of the aggregate uncontrolled dynamics \ref{['eq:vector_model']} (left) and the corresponding solution under the MPC algorithm \ref{['alg:1']} (right).
  • Figure 3: Numerical simulation of uncontrolled dynamics \ref{['eq:vector_model']} (left) and the corresponding solution under the MPC algorithm \ref{['alg:1']} (right) in each community.
  • Figure 4: Control cost vs effectiveness for constant control policy (CCP) \ref{['eq:opt-constant']} (in red) and MPC algorithm \ref{['alg:1']} (in blue).

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Theorem 2