Power-law of Aggregate-size Spectra in Natural Systems

TL;DR

This paper investigates the universal power-law distribution of aggregate sizes in natural systems, linking geometric properties with probabilistic models to understand system form, function, and dynamics across biological and inanimate systems.

Contribution

It introduces a probabilistic model for aggregate-size distribution applicable to diverse natural systems, unifying geometric and fractal analysis methods.

Findings

Aggregate-size spectra follow a double-Pareto distribution.

Perimeter-area relationships reflect the fractal nature of systems.

Variations in fractal dimension indicate system changes.

Abstract

Patterns of animate and inanimate systems show remarkable similarities in their aggregation. One similarity is the double-Pareto distribution of the aggregate-size of system components. Different models have been developed to predict aggregates of system components. However, not many models have been developed to describe probabilistically the aggregate-size distribution of any system regardless of the intrinsic and extrinsic drivers of the aggregation process. Here we consider natural animate systems, from one of the greatest mammals - the African elephant (\textit{Loxodonta africana}) - to the \textit{Escherichia coli} bacteria, and natural inanimate systems in river basins. Considering aggregates as islands and their perimeter as a curve mirroring the sculpting network of the system, the probability of exceedence of the drainage area, and the Hack's law are shown to be the the Kor\v{c}ak's law and the perimeter-area relationship for river basins. The perimeter-area relationship, and the probability of exceedence of the aggregate-size provide a meaningful estimate of the same fractal dimension. Systems aggregate because of the influence exerted by a physical or processes network within the system domain. The aggregate-size distribution is accurately derived using the null-method of box-counting on the occurrences of system components. The importance of the aggregate-size spectrum relies on its ability to reveal system form, function, and dynamics also as a function of other coupled systems. Variations of the fractal dimension and of the aggregate-size distribution are related to changes of systems that are meaningful to monitor because potentially critical for these systems.