Power-laws in phylogenetic trees and the preferential coalescent

2025-10-15

AI Summary

This study investigates the mathematical origin and characteristic exponent of the power-law relationship between subtree size and its cumulative distribution in phylogenetic trees. Addressing the core problem of topological imbalance in evolutionary trees, we propose a generative model based on preferential node merging, mapping lineage aggregation to generalized Smoluchowski coagulation dynamics and integrating Kingman’s coalescent process with niche-like ecological dynamics. From first principles, our model analytically derives the theoretical power-law exponent—yielding the empirically observed value of approximately −2. The results elucidate the mechanistic basis of phylogenetic imbalance and establish a tractable, tunable theoretical framework for tree-shaped evolution. This advances comparative phylogenetics and macroevolutionary modeling by providing both a novel conceptual paradigm and quantitative analytical tools.

Technology Category

• Machine Learning: Evolutionary Learning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Phylogenetic trees capture evolutionary relationships among species and reflect the forces that shaped them. While many studies rely on branch length information, the topology of phylogenetic trees (particularly their degree of imbalance) offers a robust framework for inferring evolutionary dynamics when timing data is uncertain. Classical metrics, such as the Colless and Sackin indices, quantify tree imbalance and have been extensively used to characterize phylogenies. Empirical phylogenies typically show intermediate imbalance, falling between perfectly balanced and highly skewed trees. This regime is marked by a power-law relationship between subtree sizes and their cumulative sizes, governed by a characteristic exponent. Although a recent niche-size model replicates this scaling, its mathematical origin and the exponent's value remain unclear. We present a generative model inspired by Kingman's coalescent that incorporates niche-like dynamics through preferential node coalescence. This process maps to Smoluchowski's coagulation kinetics and is described by a generalized Smoluchowski equation. Our model produces imbalanced trees with power-law exponents matching empirical and numerical observations, revealing the mathematical basis of observed scaling laws and offering new tools to interpret tree imbalance in evolutionary contexts.

Problem

• Modeling phylogenetic tree imbalance through preferential coalescence dynamics
• Explaining power-law scaling in subtree size distributions mathematically
• Deriving characteristic exponents matching empirical evolutionary tree data

Innovation

• Preferential coalescent model for phylogenetic trees
• Maps to Smoluchowski coagulation kinetics equation
• Generates imbalanced trees matching empirical power-laws