  We introduce a new directed graph model for social networks, based on the
transitivity of triads. In the Iterated Local Directed Transitivity (ILDT)
model, new nodes are born over discrete time-steps, and inherit the link
structure of their parent nodes. The ILDT model may be viewed as a directed
analogue of the ILT model for undirected graphs introduced in \cite{ilt}. We
investigate network science and graph theoretical properties of ILDT digraphs.
We prove that the ILDT model exhibits a densification power law, so that the
digraphs generated by the models densify over time. The number of directed
triads are investigated, and counts are given of the number of directed
3-cycles and transitive $3$-cycles. A higher number of transitive 3-cycles are
generated by the ILDT model, as found in real-world, on-line social networks.
In many instances of the chosen initial digraph, the model eventually generates
graphs with Hamiltonian directed cycles. We finish with a discussion of the
eigenvalues of the adjacency matrices of ILDT directed graphs, and provide
further directions.
