Fractal Structure with a Typical Scale

TL;DR

This paper introduces the concept of a typical scale to unify the understanding of distributions with both fractal and non-fractal regions, using a modified 2d gravity model to fit real-world data like income and citation distributions.

Contribution

It proposes a new framework incorporating a typical scale into scale-invariant models, demonstrated through an $R^2$ 2d gravity model fitting real-world data.

Findings

Distributions with fractal and non-fractal regions can be modeled using the $R^2$ 2d gravity framework.

The model fits empirical data such as personal income and citation distributions well.

The concept of a typical scale helps systematically analyze various complex distributions.

Abstract

In order to understand characteristics common to distributions which have both fractal and non-fractal scale regions in a unified framework, we introduce a concept of typical scale. We employ a model of 2d gravity modified by the $R^2$ term as a tool to understand such distributions through the typical scale. This model is obtained by adding an interaction term with a typical scale to a scale invariant system. A distribution derived in the model provides power law one in the large scale region, but Weibull-like one in the small scale region. As examples of distributions which have both fractal and non-fractal regions, we take those of personal income and citation number of scientific papers. We show that these distributions are fitted fairly well by the distribution curves derived analytically in the $R^2$ 2d gravity model. As a result, we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way. We also point out that the $R^2$ 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way.