Applied topology for phylogenetic networks

February 19, 2019
In mathematical phylogenetics we are interested in developing mathematical methods that model the structure of evolutionary relationships among biological organisms known as phylogenetic networks.  Such models are often used to define metrics for comparison, that can be used for developing statistics and probability distributions on these structures. In practice we often encounter tree-like relationships called phylogenetic trees whose structure is already very well studied.  With the cophenetic map (Sokal and Rohlf, 1962) the isomorphism classes of phylogenetic trees with n leaves embed as vectors in the [n(n+1)/2]-dimensional Euclidean space. There we pull back l^p norms to define metrics on these trees. However quite few is known for the general case of phylogenetic networks. In this talk I will introduce some concepts from applied topology, namely categories, coproducts and Reeb graphs. I will discuss why Reeb graphs is the natural model for phylogenetic networks.  We show that any phylogenetic network with n-labelled leaves and s cycles has a canonical decomposition as a join of a set of phylogenetic trees with (n + s)-labelled leaves. By combining this tree-decomposition and the cophenetic map phylogenetic networks with n-labelled leave s cycles embed as point-clouds in the [(n+s)(n+s+1)/2]-dimensional Euclidean space. Thinking of phylogenetic networks as point-clouds we can study topological signatures such as  persistence diagrams.