Some axiomatic limitations for consensus and supertree functions on hierarchies
Introduction
Motivated by the recent note of Steel and Velasco (2014) we investigate consequences of imposing a seemingly innocuous condition: removing or adding a phylogenetic tree that contains no branching information should not influence the consensus of a given set of trees. As was done in Steel and Velasco (2014), we also consider what happens when “independence” conditions like those found in Barthélemy et al. (1995) are also imposed on consensus and supertree functions on phylogenetic trees. The type of tree we are concerned with is called a hierarchy, whose precise definition follows in the next section. Informally, consensus and supertree functions can be described as follows: given a collection of hierarchies, each defined on a set of taxa S, a consensus function will return a hierarchy on S that is meant to represent a summary or agreement of the original input hierarchies. When given a collection of hierarchies, each defined on a possibly different subset of S, a supertree function will return a full hierarchy on all of S. Consensus functions have been used regularly in systematic biology; one of the first ones is due to Adams (1971). Beginning in the early 1980s the study of various consensus functions on hierarchies has been an active area of research. The idea of a supertree function was first proposed by Gordon (1986) and has become an useful tool in, for example, inferring species trees from gene trees, and construction of large phylogenies from smaller ones in the search for the Tree of Life (Bininda-Emonds, 2004).
Our contribution is to the axiomatic method, whereby reasonable and intuitive properties (axioms) are proposed for consensus and supertree functions to see what classes of functions, if any, can be constructed to satisfy various suites of axioms. See Day and McMorris (2003) for many examples of this approach up to 2003, modeled after the successful application of the axiomatic approach in social choice theory (see Aleskerov, 1999, Arrow, 1951, Kelly, 1978). Among the interesting and important companion volumes to the pure axiomatic approach are Huson et al. (2010), Semple and Steel (2003), and Dress et al. (2012). In a recent paper (Domenach and Tayari, 2016) open source software is announced that allows users to give a set of properties desired for a consensus function and then be guided to an appropriate one, if possible.
In addition to new results in the following sections, we will review some earlier related material of ours.
Section snippets
Preliminaries and definitions
Let S be a finite set with n elements. A hierarchy on S is a collection H of nonempty subsets of S (i.e., a hypergraph) such that , for all , and for all . (Note that we allow the trivial cases where n=1 or n=2.) For example, when S is a set of biological entities such as homologous molecular sequences from different taxa, then hierarchies on S could be the end result of using one of the many phylogenetic tree building algorithms based on similarity or dissimilarity
Some axioms and results for consensus functions
As mentioned in the Introduction, we are concerned in this paper with the purely axiomatic considerations in order to explore some of the limitations that might exist when reasonable axioms are imposed on consensus and supertree functions. These limitations, often more severe than anticipated, serve as important cautionary notes for researchers using consensus and supertree methods. For example, suppose a method is constructed that the inventor intuitively thinks satisfies certain properties,
Axioms and results for supertree functions
Many axioms for consensus functions have direct extensions to the supertree case. If is such an axiom, we will let be a supertree axiom such that when is applied to consensus functions only, it is exactly . Now we make an important observation that yields results that can be mechanically translated from the consensus case to the supertree case. Observation 15 Suppose it can be proved that there is no consensus function that satisfies a finite list of axioms , each one with an
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