Quartet-based species tree methods enable fast and consistent tree of blobs reconstruction under the network...

Hybridization between species is an important force in evolution, commonly modeled by the network multispecies coalescent. Reconstructing evolutionary histories under this model is computationally challenging, even for level-1 networks where hybridization events are isolated. Divide-and-conquer is a promising path forward, but current methods with statistical guarantees rely on an estimated tree of blobs (TOB) for the network, which compresses each nontree-like part into a single vertex. TOB reconstruction is itself challenging, with the only available method TINNiK having time complexity O(n ⁵ + n ⁴ k) for k genes and n species. Here, we present a new framework for scalable TOB reconstruction with statistical guarantees. Our approach operates by (1) seeking a refinement of the TOB and then (2) contracting edges in it. For step (1), we show that any optimal solution to Weighted Quartet Consensus is a TOB refinement almost surely, as the number of genes goes to infinity, motivating the use of methods, such as ASTRAL or TREE-QMC. For step (2), we show that applying the same hypothesis tests as TINNiK to just O(n) four-taxon subsets around each edge is sufficient for statistically consistent TOB reconstruction when the underlying network is level-1. Leveraging TREE-QMC for the first step gives our method time complexity O(n ³ k) and its name: TOB-QMC. On simulated data, TOB-QMC typically matches or exceeds TINNiK in accuracy while being more scalable. TOB-QMC also enables fast exploration of nontreelike evolution, as demonstrated through reanalysis of three phylogenomic data sets. Lastly, our study clarifies the theoretical utility of quartet-based species tree methods in the context of hybridization, which is critical given the recent result that ASTRAL can be misleading.