The fractal geometry of opinion formation

TL;DR

This work analyzes a stochastic opinion-dynamics model with anticonformity on $[-1,1]$, deriving its mean-field limit as $N\to\infty$ and obtaining a Boltzmann-type PDE for the opinion distribution. It proves propagation of chaos and characterizes a nonlinear mean-field process $Z_t$ whose mean $m_t$ evolves autonomously, leading to a nontrivial equilibrium $\rho_\infty$; in several parameter regimes, $\rho_\infty$ exhibits fractal structure akin to Bernoulli convolutions, with a Cantor-like support when $\mu_{+}+\mu_{-}>1$ and a Hausdorff dimension $D$ solving $(1-\mu_{+})^D+(1-\mu_{-})^D=1$. In the symmetric case $\mu_{+}=\mu_{-}$, the results reproduce the Bernoulli convolution, while the asymmetric case yields a generalized Bernoulli convolution, supported by numerical evidence and conjectures. The paper delivers quantitative convergence bounds in Wasserstein and Toscani-type metrics, including uniform-in-time results when $\mu_{+}=\mu_{-}$, and it discusses the implications for opinion fragmentation and fractal geometry in social dynamics. Overall, the work connects kinetic theory, multi-agent systems, and fractal analysis, offering a framework for understanding how collective opinion can converge to intricate, self-similar distributions.

Abstract

In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [8], within the framework of an interacting multi-agent system, where agents are assumed to interact with each other and update their opinions after each pairwise encounter. Specifically, our opinion model involves a large crowd of $N$ indistinguishable agents, each characterized by an opinion value ranging within the interval $[-1,1]$. At each update time, two agents are picked uniformly at random and the opinion of one agent will either shift by a proportion $μ_+ \in (0,1]$ towards $+1$, or by a proportion $μ_- \in (0,1]$ towards $-1$, with probabilities depending on the other agent's opinion. We rigorously derive the mean-field limit PDE that governs the large-population limit of the agent-based model and present several quantitative results demonstrating convergence to the unique equilibrium distribution. Remarkably, for a suitable choice of model parameters, the long-term equilibrium opinion profile displays a striking self-similar structure that generalizes the celebrated Bernoulli convolution, a topic extensively studied in the context of fractal geometry [23,44]. These findings also enhance our understanding of the opinion fragmentation phenomenon and may provide valuable insights for the development of more sophisticated models in future research.

Figures (3)

• Figure 1: Schematic illustration of the limiting procedure carried out for the study the multi-agent opinion dynamics \ref{['eq:dynamics']}. We begin by establishing a propagation of chaos result in the large-population regime as $N \to \infty$, wherein individual interactions between agents become negligible, giving rise to a mean-field description governed by a Boltzmann-type PDE. Once this limit is formalized, we derive a series of quantitative estimates that substantiate the convergence of the mean-field PDE solution toward its unique equilibrium distribution.
• Figure 2: Left: Monte Carlo simulation of the agent-based opinion model with $N = 5\cdot 10^6$ agents and time $t=10$, for three pairs of values of $(\mu_-, \mu_+)$ such that $\mu_- + \mu_+ > 1$. In each plot the (empirical) cumulative distribution function (CDF) is displayed, which is a good approximation of the CDF of $\rho_\infty$. Right: Evolution of the CDF, computed by solving numerically the mean-field PDE \ref{['eq:PDE']}, for the same three pairs $(\mu_-, \mu_+)$. The plots exhibit a fractal structure of $\operatorname{supp}(\rho_\infty)$, as predicted by Theorem \ref{['thm:hausdorff']}.
• Figure 3: Monte Carlo simulation of the agent-based opinion model with $N = 5\cdot 10^6$ agents and time $t=20$, for six pairs of values of $(\mu_-, \mu_+)$. In each plot the normalized histogram is displayed, as an approximation of the density of $\rho_\infty$, if it exists. Left: three pairs of values $(\mu_-, \mu_+)$ such that $\mu_- + \mu_+ < 1$. Right: three pairs of values $(\mu_-, \mu_+)$ such that $\mu_- + \mu_+ = 1$.

Theorems & Definitions (16)

• Theorem 1: Propagation of chaos
• Proposition 2
• Lemma 3
• Corollary 4
• Theorem 5: Uniform propagation of chaos when $\mu_{+} = \mu_{-}$
• Proposition 6
• Lemma 7
• Proposition 8: Weak contractivity of $Q_+$ in $W_1$ when $\mu_+ = \mu_-$
• Proposition 9: Weak contractivity of $Q_+$ in $W_2$ when $\mu_+ = \mu_-$
• Theorem 10