Upper-Truncated Power Laws and Limits to Scale Invariance in Natural Systems

In pioneering work, Mandelbrot demonstrated that many natural phenomena, from topography to galaxies, exhibit scale invariance over several orders of magnitude. Mandelbrot's fractal geometry quantifies scale invariance with a fractal dimension. The cumulative number-size distribution of a scale invariant system follows a power law where the scaling exponent is related to the fractal dimension. Many distributions that describe natural systems exhibit fall-off from a power law at the largest sizes. Previous work has often either ignored the fall-off region or described this region with a different function. Fall-off from a power law is expected for the cumulative distribution when a data set is abruptly truncated at large object size. The function that describes this truncated distribution is an upper-truncated power law. Examples of cumulative distributions that are well-described by an upper-truncated power law include fault offsets, fault lengths, forest fire areas, hydrocarbon accumulation volumes, hotspot seamount volumes, and shoreline erosion and accretion distances. For all of these examples, there are upper limits to the scale invariance of the observations. These limits may be due to spatial or temporal sampling limitations or physical limitations of the system. The scaling exponent determined from the upper-truncated power law may provide a link between observed cumulative distributions and fractal geometry within each natural system.