Combinatorial Topological Models for Phylogenetic Networks and the Mergegram Invariant

In real world, mutations of genetic sequences are often accompanied by their recombinations. Such joint phenomena are modeled by phylogenetic networks. Nakkleh formulated the phylogenetic network reconstruction problem (PNRP) as follows: Given a family of phylogenetic trees over a common set of taxa, is there a unique minimal phylogenetic network whose set of spanning trees contains the family? There are different answers to PNRP, since there are different ways to define what a minimal network is (based on different optimization criteria). Inspired by ideas from topological data analysis (TDA), we devise lattice-diagram models for the visualization of phylogenetic networks and of filtrations, called the cliquegram and the facegram, respectively, both generalizing the dendrogram model of phylogenetic trees. Both models allow us to solve the PNRP in a rigorous way and free of choosing optimization criteria. The solution to the phylogenetic network and filtration reconstruction process is obtained by taking the join operation of the dendrograms on the lattice of cliquegrams, and of facegrams, respectively. Furthermore, we show that computing the join-facegram from a given set of dendrograms is polynomial in the size and number of the input trees. We propose two novel invariants of facegrams, (i) the face-Reeb graph and (ii) the mergegram of a facegram. We show the mergegram is 1-Lipschitz stable, while the face-Reeb graph is not. In particular, we show that the mergegram is invariant of weak equivalences of filtrations (a stronger form of homotopy equivalence). This new TDA-signature, the mergegram, can be used as a computable proxy for phylogenetic networks and also, more broadly, for filtrations of datasets, which might be of independent interest to TDA. To illustrate the utility of those new TDA-tools to phylogenetics, we provide experiments with artificial and benchmark biological data.