  We derive a Belief-Propagation algorithm for counting large loops in a
directed network. We evaluate the distribution of the number of small loops in
a directed random network with given degree sequence. We apply the algorithm to
a few characteristic directed networks of various network sizes and loop
structures and compare the algorithm with exhaustive counting results when
possible. The algorithm is adequate in estimating loop counts for large
directed networks and can be used to compare the loop structure of directed
networks and their randomized counterparts.
