  We consider spectral methods that uncover hidden structures in directed
networks. We establish and exploit connections between node reordering via (a)
minimizing an objective function and (b) maximizing the likelihood of a random
graph model. We focus on two existing spectral approaches that build and
analyse Laplacian-style matrices via the minimization of frustration and
trophic incoherence. These algorithms aim to reveal directed periodic and
linear hierarchies, respectively. We show that reordering nodes using the two
algorithms, or mapping them onto a specified lattice, is associated with new
classes of directed random graph models. Using this random graph setting, we
are able to compare the two algorithms on a given network and quantify which
structure is more likely to be present. We illustrate the approach on synthetic
and real networks, and discuss practical implementation issues.
