Identifying Circular Orders for Blobs in Phylogenetic Networks

Interest in the inference of evolutionary networks relating species orpopulations has grown with the increasing recognition of the importance ofhybridization, gene flow and admixture, and the availability of large-scalegenomic data. However, what network features may be validly inferred fromvarious data types under different models remains poorly understood. Previouswork has largely focused on level-1 networks, in which reticulation events arewell separated, and on a general network's tree of blobs, the tree obtained bycontracting every blob to a node. An open question is the identifiability ofthe topology of a blob of unknown level. We consider the identifiability of thecircular order in which subnetworks attach to a blob, first proving that thisorder is well-defined for outer-labeled planar blobs. For this class of blobs,we show that the circular order information from 4-taxon subnetworks identifiesthe full circular order of the blob. Similarly, the circular order from 3-taxonrooted subnetworks identifies the full circular order of a rooted blob. We thenshow that subnetwork circular information is identifiable from certain datatypes and evolutionary models. This provides a general positive result forhigh-level networks, on the identifiability of the ordering in which taxonblocks attach to blobs in outer-labeled planar networks. Finally, we giveexamples of blobs with different internal structures which cannot bedistinguished under many models and data types.