Taxonomy of Scaling Behaviors

• Taxonomy of scaling behaviors is a systematic classification of how observables in complex systems change with size, using mathematical forms like power laws.
• It employs stochastic branching models to connect evolvability and robustness, differentiating balanced and imbalanced phylogenetic trees.
• This framework uncovers universal scaling laws across biological systems, aiding researchers in analyzing and interpreting diversification dynamics.

A taxonomy of scaling behaviors provides a systematic classification of how observables, processes, or structures in complex systems change as a function of system size, resource, or other intrinsic parameters. In quantitative terms, scaling behavior is described by mathematical forms—often but not exclusively power laws—that characterize the dependence of system properties on size, complexity, or hierarchical level. This concept plays a central role in disciplines ranging from evolutionary biology to statistical physics, network science, and information theory, as scaling behaviors often signal underlying universal principles or organizing constraints in complex systems.

1. Mathematical Characterization of Scaling Behaviors

Scaling behaviors are primarily identified through the relationship between a system's observable property and a characteristic parameter, often the size or number of components. In the context of evolutionary phylogenies, the key measure is the mean depth of a phylogenetic tree as a function of its size (number of leaves or nodes). If $C = \sum_{j} [d_\text{root,j} + 1]$ represents the cumulative branch size (sum of the topological distances from root to all nodes plus one), the mean depth is defined as: $d = \frac{C}{A} - 1$ where $A$ is the total number of nodes.

Limiting scaling behaviors are distinguished as follows:

• Logarithmic scaling (Balanced Trees): For a fully symmetric, balanced binary tree, mean depth grows as $d \sim \ln A$.
• Linear scaling (Imbalanced Trees): For a fully asymmetric, imbalanced binary tree, $d \sim A$.
• Minimal (Polytomic) Scaling: For a fully polytomic tree, $d_{\min} = 1 - 1/A$.
• Maximal Depth: For fully imbalanced trees, for large $A$, $d_{\max} \approx A$.

Empirical studies of protein and species phylogenies reveal intermediate behaviors, with mean depth often following a squared logarithmic scaling law, $d \sim (\ln A)^2$, at observable system sizes. This lies between the strictly balanced and imbalanced extremes, and the form of scaling acts as a quantitative diagnostic of phylogenetic tree topology and "balance."

2. Empirical Universality and Classification Across Biological Levels

Analysis of large datasets (e.g., PANDIT for protein families and TreeBASE for species phylogenies) demonstrates that both protein and species evolutionary trees exhibit remarkably similar depth scaling when mean depth is plotted as a function of tree size. This empirical universality persists across proteins of different functional classes, further reinforcing the notion that a common branching mechanism underlies biological diversification at genetic and organismal levels.

The taxonomy of scaling behaviors in this context is thus defined by:

• Consistency of scaling forms (e.g., $(\ln A)^2$ scaling) across levels of biological organization.
• Boundedness between minimal (balanced) and maximal (imbalanced) theoretical scaling lines.
• Quantification of departures (or convergence) towards universal scaling curves depending on evolutionary constraints and system-specific attributes.

3. Stochastic Modeling: Evolvability–Robustness Framework

To account for the observed scaling, a generative, stochastic branching model is formulated, incorporating the concepts of evolvability (capacity for diversification) and robustness (propensity to lose capacity to further diversify):

• Symmetric branching: With probability $p$, both daughter lineages inherit the potential to diversify (fully evolvable).
• Asymmetric branching (robust outcome): With probability $1 - p$, only one daughter retains evolvability, while the other is "locked" and cannot diversify further.

The model predicts the expected number of nodes after $n$ branching events as: $$A = 1 + 2\frac{z^n - 1}{z - 1} \quad \text{where } z = 1 + p$$ The cumulative branch size and mean depth then follow: $C \sim n z^n$

$d = \frac{C}{A} - 1 \sim \ln A$

This framework connects the relative rates of evolvability and robustness to the observed scaling, situating empirical trees along a continuum determined by $p$. Simulation with $p = 0.24$ reproduces the transient $(\ln A)^2$ scaling observed in data.

4. Implications for the Taxonomy of Scaling Behaviors in Evolution

The convergence of empirical phylogenies toward a universal scaling law implies that the statistical rules limiting or enabling lineage diversification—encoded by the interplay of evolvability and robustness—transcend molecular and species levels. This universality suggests the existence of fundamental constraints, such as:

• Microevolutionary-macroevolutionary connection: Processes at the gene/protein level (e.g., duplication, functional divergence) and those at the species level (speciation, extinction) reflect similar topological constraints and branching statistics.
• Unifying description: Differences among protein function classes or across biological organization scales do not significantly affect the scaling form, pointing toward a single taxonomy of diversification dynamics.

Mechanisms that curtail continuous diversification (robustness) are as crucial in shaping tree structure as those that promote adaptability (evolvability).

5. Role of Robustness and Diversification Constraints

Robustness is formally operationalized within the stochastic model as the asymmetry in branching—those lineages that become unable to further diversify, corresponding to the "locking" of evolutionary routes. The proportion of asymmetric events critically determines the tree's balance and depth scaling:

• High robustness (low $p$): Greater tendency toward imbalanced, linear scaling.
• High evolvability (high $p$): Tendency toward balanced, logarithmic scaling.

Empirical phylogenetic trees, which display intermediate scaling ($d \sim (\ln A)^2$), suggest that both robustness and evolvability are quantitatively significant and act together as system-level biological constraints.

6. Integrative Perspective and Taxonomic Synthesis

From the comparative analysis and stochastic modeling arises a comprehensive taxonomy of scaling behaviors in phylogeny, characterized by:

• Quantitative measures (e.g., mean depth, cumulative branch size) sensitive to evolutionary history and constraints.
• Generality across genetic and species phylogenies, with scaling laws reflecting universal branching processes.
• Explicit modeling of balance (tree topology) via parameters quantifying robustness and evolvability, which map empirical observations to theoretical limits.

This taxonomy provides a rigorous framework for classifying and interpreting the scaling properties of phylogenetic trees and, by extension, for understanding the statistical foundations of biological diversification. The presence or absence of universal scaling, as well as deviations from theoretical curves, serve as markers for identifying underlying mechanisms or constraints operating at various biological levels.