Revisiting scale invariance and scaling in ecology: River fractals as an example

Abstract

Scale invariance, which refers to the preservation of geometric properties regardless of observation scale, is a prevalent phenomenon in ecological systems. This concept is closely associated with fractals, and river networks serve as prime examples of fractal systems. Quantifying river network complexity is crucial for unveiling the role of river fractals in riverine ecological dynamics, and researchers have used a metric of “branching probability” to do so. Previous studies showed that this metric reflects the fractal nature of river networks. However, a recent article by Carraro and Altermatt (2022) contradicted this classical observation and concluded that branching probability is “scale dependent.” I dispute this claim and argue that their major conclusion is derived merely from their misconception of scale invariance. Their analysis in the original article (fig. 3a) provided evidence that branching probability is scale‐invariant (i.e., branching probability exhibits a power‐law scaling), although the authors erroneously interpreted this result as a sign of scale dependence. In this article, I re‐introduce the definition of scale invariance and show that branching probability meets this definition. This provided an opportunity to address the divergent use of “scale invariance” and “scaling” between fractal theory and ecology.