  We develop a method to infer community structure in directed networks where
the groups are ordered in a latent one-dimensional hierarchy that determines
the preferred edge direction. Our nonparametric Bayesian approach is based on a
modification of the stochastic block model (SBM), which can take advantage of
rank alignment and coherence to produce parsimonious descriptions of networks
that combine ordered hierarchies with arbitrary mixing patterns between groups.
Since our model also includes directed degree correction, we can use it to
distinguish non-local hierarchical structure from local in- and out-degree
imbalance -- thus removing a source of conflation present in most ranking
methods. We also demonstrate how we can reliably compare with the results
obtained with the unordered SBM variant to determine whether a hierarchical
ordering is statistically warranted in the first place. We illustrate the
application of our method on a wide variety of empirical networks across
several domains.
