  In this paper, we employ the decomposition of a directed network as an
undirected graph plus its associated node metadata to characterise the cyclic
structure found in directed networks by finding a Minimal Cycle Basis of the
undirected graph and augment its components with direction information. We show
that only four classes of directed cycles exist, and that they can be fully
distinguished by the organisation and number of source-sink node pairs and
their antichain structure. We are particularly interested in Directed Acyclic
Graphs and introduce a set of metrics that characterise the Minimal Cycle Basis
using the Directed Acyclic Graphs metadata information. In particular, we
numerically show that Transitive Reduction stabilises the properties of Minimal
Cycle Bases measured by the metrics we introduced while retaining key
properties of the Directed Acyclic Graph. This makes the metrics consistent
characterisation of Directed Acyclic Graphs and the systems they represent. We
measure the characteristics of the Minimal Cycle Bases of four models of
Transitively Reduced Directed Acyclic Graphs and show that the metrics
introduced are able to distinguish the models and are sensitive to their
generating mechanisms.
