Discriminative Measures for Comparison of Phylogenetic Trees

TL;DR

This paper introduces three new measures for comparing phylogenetic trees, offering efficient, bounded, and computable alternatives to existing metrics, with potential applications in evolutionary biology.

Contribution

The paper presents three novel dissimilarity measures for phylogenetic trees, including an efficient approximation to NNI distance and formulas with desirable mathematical properties.

Findings

d_{nav} approximates NNI distance efficiently

d_{CM} is a simple count-based dissimilarity

All measures are computable in O(n^2) time

Abstract

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of "edit length". At the same time, a closed form formula for $d_{nav}$ presents it as a weighted count of pairwise incompatibilities between clusters, lending it the character of an edge dissimilarity measure as well. A relaxation of this formula to a simple count yields another measure on all trees --- the crossing dissimilarity $d_{CM}$. Both dissimilarities are symmetric and positive definite (vanish only between identical trees) on binary hierarchies but they fail to satisfy the triangle inequality. Nevertheless, both are bounded below by the widely used Robinson-Foulds metric and bounded above by a closely related true metric, the cluster-cardinality metric $d_{CC}$. We show that each of the three proposed new dissimilarities is computable in time $O(n^2)$ in the number of leaves $n$, and conclude the paper with a brief numerical exploration of the distribution over tree space of these dissimilarities in comparison with the Robinson-Foulds metric and the more recently introduced matching-split distance.