Phylogenetic networks based on the molecular clock hypothesis

A classical result in phylogenetic trees is that a binary phylogenetic tree adhering to the molecular clock hypothesis exists if and only if the matrix of distances between taxa is ultrametric. The ultrametric condition is very restrictive. In this paper we study phylogenetic networks that can be constructed assuming the molecular clock hypothesis. We characterize distance matrices that admit such networks for 3 and 4 taxa. We also design two algorithms for constructing networks optimizing the least-squares fit.