Order distances and split systems

TL;DR

This paper explores how order distances derived from general $l_1$-distances, generated by split systems, relate to properties of the underlying distances and their associated split systems, extending known results from treelike distances.

Contribution

It generalizes the study of order distances from treelike to more general $l_1$-distances, analyzing how split system properties influence order distance characteristics.

Findings

Properties of split systems can infer features of order distances.

Order distances for $l_1$-distances relate to the structure of the underlying split systems.

Extension of treelike distance results to more general $l_1$-distances.

Abstract

Given a distance $D$ on a finite set $X$ with $n$ elements, it is interesting to understand how the ranking $R_x = z_1,z_2,\dots,z_n$ obtained by ordering the elements in $X$ according to increasing distance $D(x,z_i)$ from $x$, varies with different choices of $x \in X$. The order distance $O_{p,q}(D)$ is a distance on $X$ associated to $D$ which quantifies these variations, where $q \geq \frac{p}{2} > 0$ are parameters that control how ties in the rankings are handled. The order distance $O_{p,q}(D)$ of a distance $D$ has been intensively studied in case $D$ is a treelike distance (that is, $D$ arises as the shortest path distances in an edge-weighted tree with leaves labeled by $X$), but relatively little is known about properties of $O_{p,q}(D)$ for general $D$. In this paper we study the order distance for various types of distances that naturally generalize treelike distances in that they can be generated by split systems, i.e. they are examples of so-called $l_1$-distances. In particular we show how and to what extent properties of the split systems associated to the distances $D$ that we study can be used to infer properties of $O_{p,q}(D)$.