A stochastic model for the diffusion of competing opinions with trend-following, opposition, and indifference

TL;DR

This paper introduces a stochastic model for opinion diffusion among three agent types, capturing complex social dynamics with explicit formulas and asymptotic analysis, useful for understanding and simulating opinion formation processes.

Contribution

It presents a novel stochastic framework with explicit formulas and asymptotic results for opinion dynamics involving trend-followers, opposers, and indifferents.

Findings

Explicit formulas for opinion count moments

Asymptotic laws including LLN and CLT

Early fluctuations can persist or vanish

Abstract

We study a stochastic model for the diffusion of competing opinions in a population composed of three types of agents: trend-followers, opposers, and indifferent individuals. The decision dynamics are driven by reinforcement mechanisms, modulated by a latent trend process, allowing us to capture realistic features such as amplification, resistance, and randomness in opinion formation. We derive explicit formulas for the finite-time moments of the opinion count vector and establish a set of strong asymptotic results, including laws of large numbers, central limit theorems, laws of the iterated logarithm, and almost sure convergence of empirical distributions. In particular, we show how early fluctuations can persist or vanish depending on the balance between reinforcement and opposition. Our analysis relies on martingale techniques and offers closed-form expressions for key quantities, providing both theoretical insights and tools for simulations or applications in social dynamics, marketing, or information diffusion.