Markov embedding of ranked unlabelled evolutionary trees and its applications

2026-04-17

AI Summary

This study addresses the computational bottleneck arising from the super-exponential growth of the state space in unlabeled ranked rooted binary phylogenetic trees, as well as the limited interpretability and distributional expressiveness of the F-matrix. To overcome these challenges, the authors introduce, for the first time, a Markov chain embedding approach that substantially compresses the state space. By integrating discrete phase-type distribution theory, the method enables efficient computation of the Fréchet mean, derivation of the joint distribution of tree balance indices, and closed-form expressions for arbitrary-order moments of the F-matrix. The resulting neutrality test demonstrates superior statistical power in simulation studies, offering a novel and effective tool for statistical inference on tree-shaped structures.

Technology Category

Machine Learning: Matrix & Tensor MethodsReasoning under Uncertainty: Graphical ModelsSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Rooted bifurcating trees are mathematical objects used to model evolutionary relationships and arise naturally in both coalescent theory and phylogenetics. Recent numerical representations of tree topologies, known as F-matrices, allow for summarizing a sample of trees via Fréchet means and provide new measures of tree balance. However, the number of ranked unlabelled trees grows super-exponentially with the number of leaves. This makes computation intensive and current methods rely on mixed integer programming and simulation-based methods. Moreover, F-matrices are difficult to interpret, and their distribution is only described in terms of first- and second-order moments under neutral branching. In this paper, we introduce a Markov chain embedding of ranked and unlabelled trees that drastically decreases the size of the state space. Leveraging this embedding, we develop an algorithm that efficiently computes all Fréchet means and use discrete phase-type theory to obtain the joint distribution of tree balance indices. We also use discrete phase-type theory to generalize previous results regarding moments of F-matrices to arbitrary order for any time homogeneous and bifurcating coalescent model. Using this framework, we construct three tests for neutrality and demonstrate their improved power compared to previous methods on simulated data.

Problem

• ranked unlabelled trees
• F-matrices
• Fréchet means
• tree balance
• coalescent model

Innovation

• Markov embedding
• ranked unlabelled trees
• Fréchet means
• discrete phase-type theory
• tree balance indices