Polyhedral Combinatorics of UPGMA Cones

TL;DR

This paper analyzes the geometric structure of UPGMA cones in phylogenetics using polyhedral combinatorics, providing explicit descriptions of their extreme rays and volume calculations for small cases.

Contribution

It introduces a novel polyhedral combinatorics framework to characterize UPGMA cones, including formulas for extreme rays and volume computations.

Findings

Closed-form for extreme rays of UPGMA cones

Computed normalized volumes for small n

Enhanced understanding of UPGMA's geometric subdivision

Abstract

Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the normalized volumes of the UPGMA cones for small n. Keywords: phylogenetic trees, polyhedral combinatorics, partition lattice