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

[Correction added on 8 April 2024 after first online publication: Keywords section was added.]

## 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.

## 1 INTRODUCTION

The term “scale invariance” refers to the preservation of geometric (or mathematical) properties across different observation scales. This intriguing characteristic entails the existence of a scaling law that facilitates the generalization of spatial and temporal patterns across a broad range of scales (Clark et al., 2021; Levin, 1992; Rodríguez-Iturbe & Rinaldo, 2001). The ecological implication of scale invariance or “fractals” has gained great interest over the past few decades (Brown et al., 2002, 2004; Hubbell, 2001; Terui et al., 2018; Terui et al., 2021), and fractal theory has played a pivotal role in understanding ecosystem dynamics (Brown et al., 2002).

Rivers form complex branching networks with fractal characteristics—“there are basins within basins within basins, all of them looking alike (Rinaldo et al., 2014).” There have been concerted efforts to investigate the role of river fractals in controlling metapopulation (Mari et al., 2014; Terui et al., 2018; Yeakel et al., 2014) and metacommunity dynamics (Anderson & Hayes, 2018; Carrara et al., 2012; Terui et al., 2021). The proper quantification of river network complexity is therefore crucial, and some researchers used a metric of “branching probability,” that is, the probability of observing a joining tributary per unit river distance (Terui et al., 2018, 2021; Yeakel et al., 2014). It has been shown that this measure reflects the fractal nature of river networks because it exhibits the sign of scale invariance (Peckham & Gupta, 1999; Rodríguez-Iturbe & Rinaldo, 2001; Terui et al., 2018).

A recent article by Carraro and Altermatt (2022) countered this classical observation and concluded that branching probability is “scale dependent.” They seriously criticized past research (Anderson & Hayes, 2018; Terui et al., 2018, 2021; Yeakel et al., 2014) for the use of branching probability as a measure of river fractals, provocatively concluding “an alleged property of such random networks (branching probability) is a scale dependent quantity that does not reflect any recognized metric of rivers' fractal character…” in the Abstract. Here, I would like to communicate my concern that their major conclusion is merely a misconception of the term “scale invariance.” Indeed, the reanalysis of their data proved that branching probability is scale invariant. In what follows, I re-introduce the definition of scale invariance and show that branching probability meets this definition.

## 2 PREMISE: THE DEFINITION OF SCALE INVARIANCE

Object type | Property |
---|---|

Scale invariant | Scaling, the lack of characteristic scale |

Scale dependent | Non-scaling, the existence of characteristic scale |

The above equation is interpreted as a sign of scale invariance because the multiplicative extension/shrink of scale $x$ by factor $\lambda $ results in the same shape of the original object $y$ but with a *different scale* (Proekt et al., 2012; i.e., the structural property or function is preserved). Hence, the observed object $y$ (= function $f\left(x\right)$) is said to be scale invariant.

A general property of this function is “scaling” (Table 1), in which the dimensional physical quantity of the object (e.g., length) changes predictably across scale $x$ (Rodríguez-Iturbe & Rinaldo, 2001).

Conversely, a “scale dependent” object possesses contrasting properties (Table 1). A square exemplifies a scale dependent object. Assuming $500$ km on a side, the perimeter ($500\times 4=2000$ km) remains constant at the divisors of $500$ ($x=1,2,4,5,\dots ,500$); otherwise, the measured perimeter becomes shorter than the true length if we disregard rulers that do not fit the square's side ($y=x\u230a\frac{500}{x}\u230b\times 4$, where $\u230a\cdot \u230b$ denotes the integer part of the division; see Figure 1). Therefore, the square has characteristic scales that uniquely identify its physical quantity. This property is called “non-scaling,” which contrasts with the property of a scale-invariant object (Rodríguez-Iturbe & Rinaldo, 2001). Although I used a square for the sake of simplicity, a more general definition of nonscaling is $f\left(\mathit{\lambda x}\right)\ne {\lambda}^{z}f\left(x\right)$. Thus, “scale dependence” and “scale invariance” are contrasting observations.

## 3 POWER-LAW SCALING OF BRANCHING PROBABILITY

Here, I reanalyze the data in Carraro and Altermatt (2022) to assess the scale invariance of branching probability. By definition, branching probability is said to be scale invariant if it follows a power law of observation scale $x$. Although the observation scale can take a variety of forms, a common metric in river geomorphology is the threshold value ${A}_{T}$ that defines the minimum watershed area of the river channel. Since ${A}_{T}$ measures stream size, we extract a subset of wider river channels as ${A}_{T}$ increases (Figure 2a,b); in other words, the resolution of the river network becomes coarser. I extracted river networks of $50$ watersheds that were used in Carraro and Altermatt with MERIT Hydro (Yamazaki et al., 2019) (pixel size: $90\phantom{\rule{0.25em}{0ex}}\mathrm{m}\times 90\phantom{\rule{0.25em}{0ex}}\mathrm{m}$).

^{2}) (${A}_{T}=1,\dots ,1000$ with an equal interval at a ${\mathrm{log}}_{10}$ scale, but confined to ${A}_{T}<\text{total watershed area}$ for small rivers), at which I estimated branching probability. Branching probability $p$ (${\mathrm{km}}^{-1}$) is the probability of observing a joining tributary per unit river distance. This quantity can be estimated from the probability distribution of link length $L$ (i.e., the length of the river channel from one confluence to another, or one confluence to the outlet/upstream terminal). Typically, the link length $L$ follows an exponential distribution, and branching probability can be calculated as a cumulative distribution function of the exponential distribution (Terui et al., 2018; Terui et al., 2021):

*θ*’ in equation 4 and in the preceding sentence.].

^{−1}’ was amended to ‘

*θ*

^{−1}’ in the preceding sentence.]. The analysis provided strong support for the power-law scaling of branching probability (Figure 2c) with the estimated scaling exponent of $z=-0.48\pm 0.003$.

## 4 PRACTICAL ISSUES

It is noteworthy that the similar power-law scaling was reported in Carraro and Altermatt (2022) using the inverse of mean link length $\theta $ (referred to as “branching ratio ${p}_{r}$” in the original article; see fig. 3a and eq. [1] in Carraro and Altermatt [2022]) [Correction added on 8 April 2024 after first online publication: the symbol ‘λ’ was amended to ‘*θ*’ in the preceding sentence.]. Nevertheless, the authors erroneously interpreted this result as a sign of “scale dependence.” In addition, they used the term “scaling” as if it describes a property of scale dependence (page 3, right column). The misuse of the terms is not a simple issue of terminology. Based on their misconception, they expanded a discussion over two pages and concluded “We therefore conclude that branching probability is a non-descriptive property of a river network, which by no means describes its inherent branching character, and depends on the observational scale” (page 3). Hence, Carraro and Altermatt (2022) boldly used their misconception to invalidate past research that correctly referred to branching probability (or a cumulative distribution function of link length) as scale invariant (Moore et al., 2015; Peckham & Gupta, 1999; Rodríguez-Iturbe & Rinaldo, 2001; Terui et al., 2018, 2021).

As evident from Equation (6), we can obtain the unique value of (dimensionless) branching probability for a given river network after proper rescaling. This rescaling technique has been used for decades to characterize the physical/biological structure of scale invariant objects (Brown et al., 2004; Rinaldo et al., 2014; Rodríguez-Iturbe & Rinaldo, 2001). In fact, previous studies used a unit scale of ${A}_{T}$ (${A}_{T}=1\phantom{\rule{0.25em}{0ex}}{\mathrm{km}}^{2}$; notice that $p\approx c$ when ${A}_{T}=1$) to approximate the rescaled branching probability (Terui et al., 2018; Terui et al., 2021). As such, Carraro and Altermatt (2022) provided no evidence that limits the use of branching probability as a measure of river network structure.

Overall, it is clear that the authors fundamentally misunderstood the mathematical definition of scale invariance. Inappropriate arguments arising from the misconception spanned from pages 2 to 4; thus, correcting this misconception compromises the substantial portion of their article.

## 5 DISTINCT TERMINOLOGY IN ECOLOGY

The misconception in Carraro and Altermatt (2022) may be rooted in the distinct use of the concerned terms in ecology. Although some ecologists follow the mathematical definitions (Brown et al., 2002, 2004; Clark et al., 2021; Hatton et al., 2015; Hubbell, 2001; Levin, 1992; Sabo et al., 2010), the majority of research uses the terms in stark contrast with fractal theory. At least, there are three types of contrast.

Clearly, scale invariance is used to describe food web properties that do not change from local to regional scales in a statistical sense; this is what is called “scale dependence” (or “non-scaling”) in fractal theory. If truly scale invariant in a fractal sense, the food web properties should change according to the power law scaling of observation scale (species richness or area).… the proportion of species per trophic level and the proportion of overlap in the consumers' diet were largely

scale-invariant(fig. 3). The proportion of basal, intermediate, and top species showedsimilar values from local to regional spatial scales… (emphasis added)

Second, ecologists use scale dependence when the observation is scale invariant. Spatial scaling of biodiversity, such as species-area relationships (SARs), exemplifies this situation. Species richness typically obeys a power-law function of area, a relationship currently known as the Arrhenius SAR (Arrhenius, 1921). This scaling feature enables us to predict species richness from small to large areas because a consistent “scaling law” exists across spatial scales. This property has contributed to the spatial design of aquatic and terrestrial protected areas (Desmet & Cowling, 2004; Neigel, 2003), which encompass more than 30 million km^{2} globally (Deguignet et al., 2014). Nevertheless, ecologists refer to such scaling relationships as scale-dependent (Chase et al., 2018; Nishizawa et al., 2022; Palmer & White, 1994) in contrast to the common terminology in fractal theory. Only a few ecologists refer to power-law SARs as scale invariant following the definition of fractal theory (e.g., Hubbell, 2001).

Lastly, ecologists use scaling when scaling is impossible. Countless numbers of ecological research use the term scaling when observations do not follow the scaling relationship defined in eq. (3) (Galiana et al., 2021, 2022; Gonzalez et al., 2020; Jarzyna & Jetz, 2018; Keil et al., 2018). For example, Keil et al. (2018) related species extirpation events to area and described the observed non-monotonic relationships as scaling; however, this is what is called “non-scaling” in fractal theory. The “ecological” use of scaling can be inferred from contexts if there is no mention to fractals or scaling theory. Yet, it is exceedingly difficult to “adjust” the meaning of the term when some researchers discuss this ecological term of scaling with explicit reference to scaling theory (e.g., Gonzalez et al., 2020), as scaling theory defines scaling in a completely opposite manner (Brown et al., 2002, 2004; Mandelbrot, 1967; Proekt et al., 2012; Rinaldo et al., 2014; Rodríguez-Iturbe & Rinaldo, 2001). Such a description is self-contradicting for those who have backgrounds in fractal/scaling theory.

The divergent use of the same terms among disciplines could be a natural outcome given the changing nature of language. One may argue that this is not a serious problem because “people are skilled at deciphering meaning from context” (Hodges, 2008). I agree, but only if the central idea of the concept remains similar. For example, the term “metapopulation” is defined differently among studies (Fronhofer et al., 2012; Hanski & Gilpin, 1991; Poos & Jackson, 2012; Terui et al., 2018), but their “operational” definitions share the core of the concept. This is not the case with scale invariance. In ecology, scale invariance (and scaling) is used in a way that completely undermines the original theoretical implications of fractals. In an extreme case, researchers describe that a power function is an observation that does not satisfy scaling laws (Havens, 1992). It is likely that Carraro and Altermatt (2022)—despite their clear focus on fractals—picked this ecological conceptualization of “scale invariance” and “scaling” to invalidate past research that followed the mathematical definition of the same terms.

## 6 CONCLUDING REMARKS

Scale invariance is not unique to rivers (Brown et al., 2002). It is observed in predator–prey interactions (Hatton et al., 2015), biological metabolism (Brown et al., 2004), food chain length (Sabo et al., 2010), and SARs (Hubbell, 2001), to name just a few. Scaling is a fundamental property of scale-invariant observations, allowing for generalizations across different scales (Clark et al., 2021; Levin, 1992). However, the meanings of key terms have diverged within the field of ecology. In contrast, the same terms in fractal theory have retained their mathematically defined nature (Equation 3 and Table 1) since their initial introduction in the 1960s (Brown et al., 2002, 2004; Mandelbrot, 1967; Proekt et al., 2012; Rinaldo et al., 2014; Rodríguez-Iturbe & Rinaldo, 2001).

The gulf between ecology and fractal theory is deep, and achieving consistent terminology may prove challenging. As such, it is highly recommended that authors clarify the context of terminology in their research to promote better understanding. Simultaneously, it is crucial to encourage ecologists to recognize the mathematical definition of scale invariance. By fostering clearer communication and understanding between the fields, we can pave the way for more productive interdisciplinary collaborations. Otherwise, the divergent terminology creates unnecessary debates that hinder scientific advancement. The disrespectful arguments presented in Carraro and Altermatt (2022) serve as an unfortunate example of the negative consequences resulting from this divergence.

## ACKNOWLEDGMENTS

I thank Ryosuke Iritani and Shota Shibasaki for their comments on the earlier version of this manuscript. This material is based upon work supported by the National Science Foundation through the Division of Environmental Biology (DEB 2015634).

## CONFLICT OF INTEREST STATEMENT

None declared.

## Open Research

# DATA AVAILABILITY STATEMENT

Codes and data are available at https://github.com/aterui/public-proj_fractal-river.