Fractal Structure with a Typical Scale and Personal Income Distribution

In order to describe fractal and non-fractal distributions in a unified framework analytically, we introduce 2-dimensional quantum gravity with R term. This model is obtained by adding an interaction term with a typical scale to a scale invariant system. In addition to power law distribution, this model has Weibull-like distribution in the scale region where the typical scale is meaningful. We apply this model to distributions of personal income and citation number of scientific papers, and show that the theoretical curves obtained in this model fit the data of these distributions very well. We also evaluate the values of typical scales of these distributions and find that they are comparable with the average values of the distributions. As a result, we point out the possibility that 2d gravity with R term can be used as a useful analysis tool to read typical scales of various distributions observed in the real world in a systematic way.