Relations between a typical scale and averages in the breaking of fractal distribution

TL;DR

This paper explores the relationship between a typical scale and averages in fractal distributions, using a modified quantum gravity model to analyze real-world income data and identify deviations from power law behavior.

Contribution

It introduces a model combining fractal and non-fractal scales and demonstrates its applicability to personal income distributions, revealing new insights into their structure.

Findings

Relations between typical scale and averages hold in income data

Identifies periods with deviations indicating bubble terms

Estimates Pareto index and gap presence using derived relations

Abstract

We study distributions which have both fractal and non-fractal scale regions by introducing a typical scale into a scale invariant system. As one of models in which distributions follow power law in the large scale region and deviate further from the power law in the smaller scale region, we employ 2-dim quantum gravity modified by the $R^2$ term. As examples of distributions in the real world which have similar property to this model, we consider those of personal income in Japan over latest twenty fiscal years. We find relations between the typical scale and several kinds of averages in this model, and observe that these relations are also valid in recent personal income distributions in Japan with sufficient accuracy. We show the existence of the fiscal years so called bubble term in which the gap has arisen in power law, by observing that the data are away from one of these relations. We confirm, therefore, that the distribution of this model has close similarity to those of personal income. In addition, we can estimate the value of Pareto index and whether a big gap exists in power law by using only these relations. As a result, we point out that the typical scale is an useful concept different from average value and that the distribution function derived in this model is an effective tool to investigate these kinds of distributions.