A survey of the monotonicity and non-contradiction of consensus methods and supertree methods

TL;DR

The paper addresses how consensus and supertree methods respond to refinement of input data and discovery of new clades in phylogenetics. By formalizing monotonicity, non-contradiction, and co-Pareto properties, it evaluates established methods (strict, loose, and MR) and two supertree approaches (MRP, MRC), revealing that strict and MR are monotonic, while others are not in general; MRC is non-contradictory whereas MRP is not. It also demonstrates that MRP and MRC are not universally future-proof to the addition of new taxa and provides a constructive demonstration that infinitely many regular, monotonic, non-contradictory methods exist beyond the classical ones. These results underscore opportunities and constraints for designing robust consensus tools in phylogenetics, and suggest caution in relying on MRP/MRC as universal consensus solutions. The work therefore advances theoretical foundations for developing new, biologically plausible consensus methods with desirable properties.

Abstract

In a recent study, Bryant, Francis and Steel investigated the concept of \enquote{future-proofing} consensus methods in phylogenetics. That is, they investigated if such methods can be robust against the introduction of additional data like added trees or new species. In the present manuscript, we analyze consensus methods under a different aspect of introducing new data, namely concerning the discovery of new clades. In evolutionary biology, often formerly unresolved clades get resolved by refined reconstruction methods or new genetic data analyses. In our manuscript we investigate which properties of consensus methods can guarantee that such new insights do not disagree with previously found consensus trees, but merely refine them, a property termed \emph{monotonicity}. Along the lines of analyzing monotonicity, we also study two {established} supertree methods, namely Matrix Representation with Parsimony (MRP) and Matrix Representation with Compatibility (MRC), which have also been suggested as consensus methods in the literature. While we (just like Bryant, Francis and Steel in their recent study) unfortunately have to conclude some negative answers concerning general consensus methods, we also state some relevant and positive results concerning the majority rule ($\mathtt{MR}$) and strict consensus methods, which are amongst the most frequently used consensus methods. Moreover, we show that there exist infinitely many consensus methods which are monotonic and have some other desirable properties. \textbf{Keywords:} consensus tree, phylogenetics, majority rule, tree refinement, matrix representation with parsimony \textbf{MSC:} C92B05, 05C05

Figures (7)

• Figure 1: All phylogenetic $X$-trees for $X=\{1,2,3,4\}$. The Newick formats of the given trees are $T_1=(1,2,3,4)$, $T_2=((1,2),3,4)$, $T_3=((1,3),2,4)$ and $T_4=((1,4),2,3)$. Here, we have $\Sigma^*(T_1)=B^*(T_1)=\emptyset$ and $\Sigma^*(T_2)=\{12 | 34\}$, and thus $B^*(T_2)=\{1100\}$, for instance. Moreover, we have $T_1\preceq T_2$, $T_1\preceq T_3$ and $T_1\preceq T_4$, but there is no such relation between $T_2$, $T_3$ and $T_4$.
• Figure 2: The Adams and Aho consensus methods are not non-contradictory as $\varphi_{Ad}(T_1,T_2)=\varphi_{Aho}(T_1,T_2)=T_3$.
• Figure 3: Trees $T_1$, $T_2$ and $T_3$ as needed for the proof of Theorem \ref{['thm:mrpnotnoncontr']}. The alignment for $\mathcal{P}=(T_1,T_2)$ is depicted in Table \ref{['tab:mrpnotnoncontr']}.
• Figure 4: Profile $\widetilde{\mathcal{P}}$, which can be turned into profile $\mathcal{P}$ by adding information on taxon 5, leading to alignments $\mathcal{A}_{\widetilde{\mathcal{P}}}$ and $\mathcal{A}_\mathcal{P}$ as depicted in Figures \ref{['fig_alignments1']} and \ref{['fig_alignments2']}, respectively. $T_1$ is the unique MRP and MRC tree for $\mathcal{P}$, while $\widetilde{T_2}$ is the unique MRP and MRC tree for $\widetilde{T}$.
• Figure 5: Alignment $\mathcal{A}_{\widetilde{\mathcal{P}}}$, which has both a unique MRP tree as well as a unique MRC tree: both of them equal $\widetilde{T_2}$ as depicted in Figure \ref{['fig_MRPMRCaddTaxon']}. The unique maximum set of compatible characters in $\mathcal{A}_{\widetilde{\mathcal{P}}}$ is hightlighted in bold.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark
• Remark
• Theorem 3.1: Theorem 1 in mcmorris2008characterization, Theorem 2 in mcmorris
• Proposition 3.2
• proof
• Proposition 3.3