Order distances and split systems
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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,...,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 ∈ X. The order distance O_p,q(D) is a distance on X associated to D which quantifies these variations, where q ≥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).
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