  Interest in the inference of evolutionary networks relating species or
populations has grown with the increasing recognition of the importance of
hybridization, gene flow and admixture, and the availability of large-scale
genomic data. However, what network features may be validly inferred from
various data types under different models remains poorly understood. Previous
work has largely focused on level-1 networks, in which reticulation events are
well separated, and on a general network's tree of blobs, the tree obtained by
contracting every blob to a node. An open question is the identifiability of
the topology of a blob of unknown level. We consider the identifiability of the
circular order in which subnetworks attach to a blob, first proving that this
order is well-defined for outer-labeled planar blobs. For this class of blobs,
we show that the circular order information from 4-taxon subnetworks identifies
the full circular order of the blob. Similarly, the circular order from 3-taxon
rooted subnetworks identifies the full circular order of a rooted blob. We then
show that subnetwork circular information is identifiable from certain data
types and evolutionary models. This provides a general positive result for
high-level networks, on the identifiability of the ordering in which taxon
blocks attach to blobs in outer-labeled planar networks. Finally, we give
examples of blobs with different internal structures which cannot be
distinguished under many models and data types.
