The Pfaffian Structure of CFN Phylogenetic Networks

Algebraic techniques in phylogenetics have historically been successful at proving identifiability results and have also led to novel reconstruction algorithms. In this paper, we study the ideal of phylogenetic invariants of the Cavender-Farris-Neyman (CFN) model on a phylogenetic network with the goal of providing a description of the invariants which is useful for network inference. It was previously shown that to characterize the invariants of any level-1 network, it suffices to understand all sunlet networks, which are those consisting of a single cycle with a leaf adjacent to each cycle vertex. We show that the parameterization of an affine open patch of the CFN sunlet model, which intersects the probability simplex, factors through the space of skew-symmetric matrices via Pfaffians. We then show that this affine patch is isomorphic to a determinantal variety and give an explicit Gr{\"o}bner basis for the associated ideal, which involves only $\binom{n}{2}$ coordinates rather than $2^{n}$. Lastly, we show that sunlet networks with at least 6 leaves are identifiable using only these polynomials and run extensive simulations, which show that these polynomials can be used to accurately infer the correct network from DNA sequence data.