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Steering Opinion Dynamics in Signed Time-Varying Networks via External Control Input

Swati Priya, Twinkle Tripathy

TL;DR

The paper addresses targeted opinion formation in networks with time-varying signed interactions using external control inputs. It models the dynamics as $\\dot{\\mathbf{x}}(t) = -L(t)\\mathbf{x}(t) + \\mathbf{u}(t)$ on graphs that are uniformly quasi-strongly $\\delta$-connected with a persistently structurally balanced root SCC. A decentralized controller $\\mathbf{u}(t) = L(t)\\mathbf{x}_d - K(t)(\\mathbf{x}(t) -\\mathbf{x}_d)$ with $K(t)$ positive on the root set drives the state to the target exponentially. The analysis combines the upper Dini derivative and Grönwall inequalities, supported by numerical simulations, to demonstrate reliable, structure-aware steering of collective opinions in evolving social networks.

Abstract

This paper studies targeted opinion formation in multi-agent systems evolving over signed, time-varying directed graphs. The dynamics of each agent's state follow a Laplacian-based update rule driven by both cooperative and antagonistic interactions in the presence of exogenous factors. We formulate these exogenous factors as external control inputs and establish a suitable controller design methodology enabling collective opinion to converge to any desired steady-state configuration, superseding the natural emergent clustering or polarization behavior imposed by persistently structurally balanced influential root nodes. Our approach leverages upper Dini derivative analysis and Grönwall-type inequalities to establish exponential convergence for opinion magnitude towards the desired steady state configuration on networks with uniform quasi-strong $δ$-connectivity. Finally, the theoretical results are validated through extensive numerical simulations.

Steering Opinion Dynamics in Signed Time-Varying Networks via External Control Input

TL;DR

The paper addresses targeted opinion formation in networks with time-varying signed interactions using external control inputs. It models the dynamics as on graphs that are uniformly quasi-strongly -connected with a persistently structurally balanced root SCC. A decentralized controller with positive on the root set drives the state to the target exponentially. The analysis combines the upper Dini derivative and Grönwall inequalities, supported by numerical simulations, to demonstrate reliable, structure-aware steering of collective opinions in evolving social networks.

Abstract

This paper studies targeted opinion formation in multi-agent systems evolving over signed, time-varying directed graphs. The dynamics of each agent's state follow a Laplacian-based update rule driven by both cooperative and antagonistic interactions in the presence of exogenous factors. We formulate these exogenous factors as external control inputs and establish a suitable controller design methodology enabling collective opinion to converge to any desired steady-state configuration, superseding the natural emergent clustering or polarization behavior imposed by persistently structurally balanced influential root nodes. Our approach leverages upper Dini derivative analysis and Grönwall-type inequalities to establish exponential convergence for opinion magnitude towards the desired steady state configuration on networks with uniform quasi-strong -connectivity. Finally, the theoretical results are validated through extensive numerical simulations.

Paper Structure

This paper contains 10 sections, 4 theorems, 41 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Notations and Preliminaries
  3. Problem statement
  4. Network dynamics and assumptions
  5. Arcs, graph, and connectivity
  6. Targeted Opinion Formation over Time-Varying Graphs
  7. Controller design methodology
  8. Numerical Simulations
  9. Discussion
  10. Conclusion

Key Result

Lemma 1

Let the time-varying interaction graph $\mathcal{G}_A(t)$ be uniformly quasi strongly $\delta$-connected such that assumptions (A1)-(A3) holds. Then for the system system_1 in the absence of control input i.e., $\mathbf{u}(t)\equiv 0$, the following dynamics holds $\forall i\in V$ and for a.a. (almo where $\theta_{ij}(t):=\operatorname{sgn}(x_i(t))\operatorname{sgn}(a_{ij}(t))\operatorname{sgn}(x_

Figures (3)

  • Figure 1: Sequence of graphs $\{\mathcal{G}_1, \mathcal{G}_2, \ldots, \mathcal{G}_5 \}$ over the interval $[0, 10)$.
  • Figure 2: Opinion evolution for system dynamics \ref{['system_1']} with $\mathbf u(t)=0$.
  • Figure 3: Desired opinion clustering of agents with control input $\mathbf{u}(t)$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more