Identifiability of local and global features of phylogenetic networks from average distances

Abstract

Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify that the root and edge lengths adjacent to reticulations are not identifiable, and then focus on the class of zipped-up semidirected networks. We provide a criterion to swap subgraphs locally, such as 3-cycles, resulting in undistinguishable networks. We propose the "distance split tree", which can be constructed from pairwise distances, and prove that it is a refinement of the network's blob tree, capturing the tree-like features of the network. For level-1 networks, this distance split tree is equal to the blob tree refined to separate polytomies from blobs, and we prove that the mixed representation of the network is identifiable. The information loss is localized around 4cycles, for which the placement of the reticulation is unidentifiable. The mixed representation combines split edges for 4-cycles, with regular tree and hybrid edges from the semidirected network, with edge parameters that encode all information identifiable from average pairwise distances.

The Tree of Blobs of a Species Network: Identifiability under the Coalescent

“…A network's blobs can equivalently be defined as the 2-edge-connected components [30], or as the connected components obtained by deleting all cut edges in the network.…”

“…An equivalent construction of a blob tree in [30] has nodes for each blob in N , with edges connecting them if there is an edge with endpoints in the two blobs in N .…”

“…Using different notions of data, however, several works have studied the identifiability question for general networks without the coalescent. Researchers have, for instance, investigated what can be determined from average intertaxon distances on a network [30], as well as shortest distances and distance multisets [29]. Identifiability from induced 4-taxon networks [16], rooted 3-taxon networks [26], and counts of paths from interior nodes to taxa [9] have also been explored, among other notions.…”