The Geometry of the Neighbor-Joining Algorithm for Small Trees

TL;DR

This paper explores the geometric structure of the Neighbor-Joining algorithm for small trees, analyzing polyhedral subdivisions and robustness to perturbations in phylogenetic reconstruction.

Contribution

It provides a detailed geometric and combinatorial analysis of NJ for five and six taxa, including hyperplane representations and robustness studies.

Findings

Polyhedral subdivision structures for six taxa analyzed

Hyperplane representations of subdivisions developed

Robustness of NJ to small perturbations studied

Abstract

In 2007, Eickmeyer et al. showed that the tree topologies outputted by the Neighbor-Joining (NJ) algorithm and the balanced minimum evolution (BME) method for phylogenetic reconstruction are each determined by a polyhedral subdivision of the space of dissimilarity maps ${\R}^{n \choose 2}$, where $n$ is the number of taxa. In this paper, we will analyze the behavior of the Neighbor-Joining algorithm on five and six taxa and study the geometry and combinatorics of the polyhedral subdivision of the space of dissimilarity maps for six taxa as well as hyperplane representations of each polyhedral subdivision. We also study simulations for one of the questions stated by Eickmeyer et al., that is, the robustness of the NJ algorithm to small perturbations of tree metrics, with tree models which are known to be hard to be reconstructed via the NJ algorithm.