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

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

7 pages, 2 figures

References in corpus (4)

• Geospatial Analysis Requires a Different Way of Thinking: The Problem of Spatial Heterogeneity
• The Evolution of Natural Cities from the Perspective of Location-Based Social Media
• Head/tail Breaks for Visualization of City Structure and Dynamics
• Wholeness as a Hierarchical Graph to Capture the Nature of Space