  In many real world networks, the number of links increases nonlinearly with
the number of nodes. Models of such accelerated growth have been considered
earlier with deterministic and stochastic number of links. Here we consider
stochastic accelerated growth in a network where links are directed. With the
number of out-going links following a power law distribution, the results are
similar to the undirected case. As the accelerated growth is enhanced, the
degree distribution becomes independent of the ``initial attractiveness'', a
parameter which plays a key role in directed networks. As an example of a
directed model with accelerated growth, the citation network is considered, in
which the distribution of the number of outgoing link has an exponential tail.
The role of accelerated growth is examined here with two different growth laws.
