Hierarchical logistic equation to describe the dynamical behavior of penetration rates

• We unveil human communication behind the logistic equation as a diffusion model.
• The essential communication is to imitate adopters easily, which is so unnatural.
• The logistic equation is improved to the hierarchical logistic equation.
• Our model reflects a fact that there exist from the trend-conscious to the cautious.
• Our model can approximate the iPod sales data better than the logistic equation.

We propose a hierarchical logistic equation as a model to describe the dynamical behavior of a penetration rate of a new product. In this model, a memory of how many adopters a non-adopter met is considered, where (non-)adopters mean people (not) possessing the product. This effect is not included in the logistic model. As an application, we apply this model to iPod unit sales data, and find that this model can approximate the data much better than the logistic equation. By using this model, we can guess a percentage of trend-cautious people in our society from the unit sales data.