Stochastic Opinion Diffusion with Trend-Following and Opposition

Summary

This document presents a stochastic model of how opinions spread through a population containing trend-followers, opposers, and indifferent people. Reinforcement drives agents’ choices, while an unobserved trend process shapes those decisions. The framework represents amplification, resistance, and randomness in opinion formation.

The analysis derives finite-time moments for the counts of each agent type and establishes asymptotic results, including laws of large numbers, central limit behavior, iterated-logarithm results, and convergence of empirical proportions. Martingale methods support the derivations, and the authors report closed-form expressions for important quantities. The document emphasizes that initial random differences may endure or fade according to the relative strength of reinforcement and opposition. It offers mathematical tools that could inform simulations of social, marketing, or information-diffusion processes, but gives no empirical market evidence or direct trading strategy.

Key ideas

• Opinion diffusion is modeled across trend-followers, opposers, and indifferent agents.
• Reinforcement and a latent trend jointly shape agents’ decisions.
• The analysis provides finite-time moments and several large-sample convergence results.
• Whether early random fluctuations persist depends on the balance between reinforcement and opposition.
• The results are theoretical and do not establish a financial-market strategy.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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