How Complex Is a Fractal? Head/tail Breaks and Fractional Hierarchy

TL;DR

This paper introduces a fractional ht-index to more accurately measure the complexity and hierarchical structure of fractals, improving upon the traditional integer-based ht-index through case studies.

Contribution

It refines the ht-index as a fractional measure and introduces the fht-index for individual data points, enhancing fractal hierarchy characterization.

Findings

Fractional ht-index provides more precise fractal measurement.

The fht-index characterizes individual data points within fractals.

Fractional hierarchy can inform intermediate map scales.

Abstract

A fractal bears a complex structure that is reflected in a scaling hierarchy, indicating that there are far more small things than large ones. This scaling hierarchy can be effectively derived using head/tail breaks - a clustering and visualization tool for data with a heavy-tailed distribution - and quantified by an ht-index, indicating the number of clusters or hierarchical levels, a head/tail breaks-induced integer. However, this integral ht-index has been found to be less precise for many fractals at their different phrases of development. This paper refines the ht-index as a fraction to measure the scaling hierarchy of a fractal more precisely within a coherent whole, and further assigns a fractional ht-index - the fht-index - to an individual data value of a data series that represents the fractal. We developed two case studies to demonstrate the advantages of the fht-index, in comparison with the ht-index. We found that the fractional ht-index or fractional hierarchy in general can help characterize a fractal set or pattern in a much more precise manner. The index may help create intermediate map scales between two consecutive map scales. Keywords: Ht-index, fractal, scaling, complexity, fht-index