Biogeographic interpretation of splits graphs: Least squares optimization of branch lengths

Although most often used to represent phylogenetic uncertainty, network methods are also potentially useful for describing the phylogenetic complexity expected to characterize recent species radiations. One network method with particular advantages in this context is split decomposition. However, in its standard implementation this approach is limited by a conservative criterion for branch length estimation. Here we extend the utility of split decomposition by introducing a least squares optimization technique for correcting branch lengths that may be underestimated by the standard implementation. This optimization of branch lengths is generally expected to improve divergence time estimates calculated from splits graphs. We illustrate the effect of least squares optimization on such estimates using the Australasian Myosotis and the Hawaiian silversword alliance as examples. We also discuss the biogeographic interpretation and limitations of splits graphs.