Scale-invariance underlying the logistic equation and its social applications

TL;DR

This paper derives the logistic equation from scale-invariance principles, generalizes it to multi-component systems, and demonstrates its applicability to various social phenomena, including city populations and web browser usage.

Contribution

It establishes scale-invariance as a fundamental principle underlying the logistic equation and extends it to multi-component systems for social applications.

Findings

Logistic equation derived from scale-invariance and mean-value constraints.

Multi-component logistic models fit city populations and network diffusion.

Predicted browser usage trends for the next 60 months.

Abstract

On the basis of dynamical principles we derive the Logistic Equation (LE), widely employed (among multiple applications) in the simulation of population growth, and demonstrate that scale-invariance and a mean-value constraint are sufficient and necessary conditions for obtaining it. We also generalize the LE to multi-component systems and show that the above dynamical mechanisms underlie large number of scale-free processes. Examples are presented regarding city-populations, diffusion in complex networks, and popularity of technological products, all of them obeying the multi-component logistic equation in an either stochastic or deterministic way. So as to assess the predictability-power of our present formalism, we advance a prediction, regarding the next 60 months, for the number of users of the three main web browsers (Explorer, Firefox and Chrome) popularly referred as "Browser Wars".