Advances in Estimating Level-1 Phylogenetic Networks from Unrooted SNPs.

We address the problem of how to estimate a phylogenetic network when given single-nucleotide polymorphisms (i.e., SNPs, or bi-allelic markers that have evolved under the infinite sites assumption). We focus on level-1 phylogenetic networks (i.e., networks where the cycles are node-disjoint), since more complex networks are unidentifiable. We provide a polynomial time quartet-based method that we prove correct for reconstructing the semi-directed level-1 phylogenetic network , if we are given a set of SNPs that covers all the bipartitions of , even if the ancestral state is not known, provided that the cycles are of length at least 5; we also prove that an algorithm developed by Dan Gusfield in the in 2005 correctly recovers semi-directed level-1 phylogenetic networks in polynomial time in this case. We present a stochastic model for DNA evolution, and we prove that the two methods (our quartet-based method and Gusfield's method) are statistically consistent estimators of the semi-directed level-1 phylogenetic network. For the case of multi-state homoplasy-free characters, we prove that our quartet-based method correctly constructs semi-directed level-1 networks under the required conditions (all cycles of length at least five), while Gusfield's algorithm cannot be used in that case. These results assume that we have access to an oracle for indicating which sites in the DNA alignment are homoplasy-free, and we show that the methods are robust, under some conditions, to oracle errors.