A Space of Phylogenetic Networks

Abstract

Abstract. A classical problem in computational biology is constructing a phylogenetic tree given a set of distances between n species. In many cases, a tree structure is too constraining. We consider a split network, which is a generalization of a tree in which multiple parallel edges signify divergence. A geometric space of such networks is introduced, forming a natural extension of the familiar space of phylogenetic trees. We explore properties of the space of networks and construct a natural embedding of the compactification of the real moduli space of curves within it.

Level-1 phylogenetic networks and their balanced minimum evolution polytopes

“…Note that these drawings are not fixed in the plane-twisting around a bridge gives a different diagram but represents the same split network. As shown in [4], there is an equivalent dual polygonal representation of any circular split network: Given a circular split system with a circular ordering c of the species, consider a regular n-gon, with the edges cyclically labeled according to c. For each split, draw a diagonal partitioning the appropriate edges; see Figure 1. Note that splits which require multiple parallel edges in the the network picture correspond to diagonals that are crossing (they intersect other diagonals in the picture) and that bridges become noncrossing diagonals.…”

Random walks and Brownian motion on cubical complexes

“…This induces a constraint on the trees, so that the path length of each leaf from the root is the same for all leaves, each of which represents a present-day species. Secondly, a space of phylogenetic networks has recently been constructed [8] which is a cubical complex. Phylogenetic networks are a generalization of phylogenetic trees that model 'non-vertical' patterns of evolution such as hybridization events.…”

“…In this way, the structure is smooth in the interior of strata, but can become singular where strata are joined together. Important examples include shape spaces [4] and, of particular relevance to this article, spaces of trees [5,6,7] and networks [8]. Cubical complexes are manifold-stratified spaces built by joining high-dimensional cubes at their faces [9,10].…”

“…Each cube is equipped with the L 2 Euclidean metric, and for certain cubical complexes this extends to define a metric on the whole complex. Cubical complexes provide a concrete way to construct stratified sample spaces, and various different spaces of evolutionary trees are constructed in this way [5,7,8]. The tree space of Billera, Holmes and Vogtmann [5], usually referred to as BHV tree space, in particular has received considerable research attention, with methods for computing Fréchet means and variances [11], constructing principal components [12], and performing other statistical tests [13].…”

The space of equidistant phylogenetic cactuses

“…Most of this paper is devoted to proving a crucial combinatorial result concerning rooted X-cactuses (Theorem 11) which implies, via a classical result of Gromov for orthant spaces, that N(X) is CAT(0). In passing, we remark that the space of networks described in [15] is not a CAT(0)metric space.…”