Metastable opinion dynamics with hidden preferences: an Ising model with neutral agents

TL;DR

This work introduces a hidden-preference Ising-type model on a toric grid that separates immutable private biases from publicly expressed binary opinions and incorporates neutral agents. Through a pathwise metastability analysis in the low-temperature limit, the authors rigorously characterize stable and metastable configurations, quantify the maximal stability barrier, and derive sharp asymptotics for hitting and mixing times; they also develop a novel geometric framework using polyominoes on the torus and new isoperimetric inequalities to identify critical configurations and energy barriers. The results reveal how spatial patterns of hidden preferences reshape collective transitions and demonstrate how geometry and probabilistic methods jointly elucidate metastability in complex social dynamics. These findings provide a rigorous quantitative understanding of regime-dependent transitions in spatially structured opinion models with neutrality and immutable private bias.

Abstract

We introduce a new Ising-type framework for opinion dynamics that explicitly separates private preferences from publicly expressed binary opinions and naturally incorporates neutral agents. Each individual is endowed with an immutable hidden preference, while public opinions evolve through Metropolis dynamics on a finite graph. This formulation extends classical sociophysical Ising models by capturing the tension between internal conviction, social conformity, and neutrality. Focusing on highly symmetric grid networks and spatially structured hidden-preference patterns, we analyze the resulting low-temperature dynamics using the pathwise approach to metastability. We provide a complete characterization of stable and metastable configurations, identify the maximal stability level of the energy landscape, and derive sharp asymptotics for hitting and mixing times. A central technical contribution is a new family of isoperimetric inequalities for polyominoes on the torus, which emerge from a geometric representation of opinion clusters and play a key role in determining critical configurations and energy barriers. Our results provide a quantitative understanding of how spatial heterogeneity in hidden preferences qualitatively reshapes collective opinion transitions and illustrate the power of geometric and probabilistic methods in the study of complex interacting systems.

Figures (10)

• Figure 2.1: Schematic representation of the $N\times N$ toric grid $G$ and the subsets $A,B$ and $C=S_1\cup S_2$, where we highlight with different colors (light blue, dark blue and green) the subsets of nodes having hidden preference $0$, $+1$ and $-1$, respectively.
• Figure 3.1: An example of Peierls contour.
• Figure 3.2: An example of the subset of sites $\mathscr{A}_{\ell,p}$, which are highlighted in dark grey, with $\ell$ and $p$ adjacent columns in $S_1$ and $S_2$, respectively.
• Figure 3.3: Representation of a configuration $\Sigma_{\mathscr{C}^A_{n-i-1,N-j}}$. Light and dark gray denote the negative and the positive region, respectively.
• Figure 3.4: On the left, a polyomino with a hole. On the right, a convex polyomino.

Theorems & Definitions (59)

• Theorem 2.1: Identification of stable configurations
• Theorem 2.2: Identification of maximum stability level and metastable states
• Theorem 2.3: Asymptotic behavior of the transition time
• Theorem 2.4: Gate for the transition from (meta)stable state to stable states
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.6