Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spatially-invariant opinion dynamics on the circle

Giovanna Amorim, Anastasia Bizyaeva, Alessio Franci, Naomi Ehrich Leonard

TL;DR

Inspired by perceptual decision-making, new theory for opinion formation in response to inputs about options distributed on the circle is developed and used to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis.

Abstract

We propose and analyze a nonlinear opinion dynamics model for an agent making decisions about a continuous distribution of options in the presence of input. Inspired by perceptual decision-making, we develop new theory for opinion formation in response to inputs about options distributed on the circle. Options on the circle can represent, e.g., the possible directions of perceived objects and resulting heading directions in planar robotic navigation problems. Interactions among options are encoded through a spatially invariant kernel, which we design to ensure that only a small (finite) subset of options can be favored over the continuum. We leverage the spatial invariance of the model linearization to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis. We illustrate our model's versatility with an application to robotic navigation in crowded spaces.

Spatially-invariant opinion dynamics on the circle

TL;DR

Inspired by perceptual decision-making, new theory for opinion formation in response to inputs about options distributed on the circle is developed and used to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis.

Abstract

We propose and analyze a nonlinear opinion dynamics model for an agent making decisions about a continuous distribution of options in the presence of input. Inspired by perceptual decision-making, we develop new theory for opinion formation in response to inputs about options distributed on the circle. Options on the circle can represent, e.g., the possible directions of perceived objects and resulting heading directions in planar robotic navigation problems. Interactions among options are encoded through a spatially invariant kernel, which we design to ensure that only a small (finite) subset of options can be favored over the continuum. We leverage the spatial invariance of the model linearization to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis. We illustrate our model's versatility with an application to robotic navigation in crowded spaces.
Paper Structure (11 sections, 4 theorems, 14 equations, 5 figures)

This paper contains 11 sections, 4 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Opinion Dynamics on the Circle
  4. Spectral Analysis of Linearization
  5. Opinion-Forming Bifurcations
  6. Decision-Making on the Circle with Tunable Sensitivity to Distributed Input
  7. Application to Robot Navigation
  8. Fourier-Based Kernel Design
  9. Choosing the Best Gap and Avoiding Deadlock
  10. Robustness and Responsiveness to Change
  11. Discussion

Key Result

Lemma 1

Define the nonlinear operator in eq:CO_DA as $[Fz](\theta,t) = \int_{\mathbb{S}^1}W(\theta - \phi)S(z(\phi,t))$. The differential of $F$ in the direction $z$ at $z(\theta,t) = 0$ is The linearization of eq:CO_DA at the neutral equilibrium $z(\theta,t) = 0$, is a spatially invariant system in the sense of Definition def:spatial_invt_sys.

Figures (5)

  • Figure 1: Bifurcation diagrams illustrating the effect of the shift value $\xi$ on the dynamics \ref{['eq:CO_DA']} with shifted sigmoid \ref{['eq:shifted_tanh']}. Stable (unstable) branches of equilibria are shown as solid (dashed) lines.
  • Figure 2: Influence of the kernel design on the steady-state opinion patterns of \ref{['eq:CO_DA']} with zero-input. (a) Two kernel designs. (b) Fourier coefficients of the kernel. Top: $\pm k_{\max} =\pm1$. Bottom: $\pm k_{\max} = \pm3$. (c) Steady-state opinion pattern $z(\theta,\infty)$, of dynamics \ref{['eq:CO_DA']} for initial conditions $z(\theta,0)$. The number of maxima of $z(\theta,\infty)$ equals $k_{\max}$ of the corresponding kernel. Parameters: $\tau = 1$, $\alpha = 0.98$, $p = 3$, $\xi = 0.7$.
  • Figure 3: Input-output behavior of the dynamics \ref{['eq:CO_DA']} with input distributions aligned or unaligned with the Fourier mode corresponding to $\pm k_{\max} \!=\! \pm1$. Top row: Aligned. Bottom row: Unaligned. (a) Magnitude of the Fourier coefficients of the input. (b) Input distribution. (c) Steady-state opinion pattern $z(0,\infty)$. Parameters: $\tau \!=\! 1$, $\alpha \!=\! 0.98$, $p \!=\! 3$, $\xi \!=\! 0.7$.
  • Figure 4: Decision-making of a robot selecting a gap through which to cross a circle of non-moving people. Bottom row: gap distribution over time where gaps are indicated by $u(\theta,t)>0$ in blue. Top row: opinion pattern over time (strongest opinion in yellow). (a) One widest gap. (b) Two wide gaps of same size. Parameters: $\tau = 1$, $\alpha = 0.98$, $\xi = 0.7$, $p = 3$.
  • Figure 5: Decision-making of a robot selecting a gap through which to cross a circle of moving people. Bottom row: gap distribution over time where gaps are indicated by $u(\theta,t)>0$ in blue. Top row: opinion about where to cross the line over time (strongest opinion in yellow). (a) Small decrease over time in size of initially widest gap. (b) Large decrease over time in size of initially widest gap. Parameters: $\tau \!=\! 1$, $\alpha \!=\! 0.96$, $\xi \!= \!0.6$, $p \!= \!3$.

Theorems & Definitions (12)

  • Definition 1: Differential operator
  • Definition 2: Multiplication Operator
  • Definition 3: Spatial shift operator bamieh2002, arbelaiz2024optimalestimationspatiallydistributed
  • Definition 4: Spatially invariant operator bamieh2002, arbelaiz2024optimalestimationspatiallydistributed
  • Definition 5: Spatially Invariant Linear System bamieh2002, arbelaiz2024optimalestimationspatiallydistributed
  • Definition 6: Spatial Fourier transform bamieh2002, arbelaiz2024optimalestimationspatiallydistributed
  • Definition 7: Temporal Laplace Transform
  • Lemma 1: Spatial invariance of the model linearization
  • Lemma 2: Eigenvalues and eigenfunctions of the linearized system
  • Lemma 3: Existence of a bifurcation point at the neutral equilibrium
  • ...and 2 more