BROOQS: Spectral Methods Resolve Level-1 Hybridization Cycles without Tests of Symmetry

Modern phylogenomic analyses often seek to reconstruct both vertical and reticulate evolutionary histories. While the prevalence of non-vertical evolution is increasingly appreciated, inferring networks remains conceptually challenging and computationally demanding. Following the success of quartet-based methods for handling gene tree discordance, several quartet-based network inference methods have been developed. A key insight of these methods is that level-1 networks can be constructed by first building a multifurcating tree called tree-of-blobs and then resolving each polytomy into a cycle. This two-step approach makes the problem easier both conceptually and computationally. However, these quartet-based methods often rely on noisy statistical tests of asymmetry in quartet frequencies. Moreover, they either enumerate all quartets, losing some scalability, or subsample them, losing information. We introduce BROOQS, a quartet-based method for resolving trees of blobs into a level-1 phylogenetic network. BROOQS efficiently aggregates information from all quartets around a blob without enumerating them, builds a pairwise similarity matrix, and uses robust spectral ordering algorithms to recover the cyclic ordering without relying on individual quartet symmetry tests. We prove theoretically that our spectral method is consistent under the network multi-species coalescent (NMSC) model. Across simulated and empirical datasets, BROOQS consistently improves accuracy and scalability compared to existing methods and extends to thousands of taxa.