  We present a linear stability analysis of stationary states (or fixed points)
in large dynamical systems defined on random directed graphs with a prescribed
distribution of indegrees and outdegrees. We obtain two remarkable results for
such dynamical systems: First, infinitely large systems on directed graphs can
be stable even when the degree distribution has unbounded support; this result
is surprising since their counterparts on nondirected graphs are unstable when
system size is large enough. Second, we show that the phase transition between
the stable and unstable phase is universal in the sense that it depends only on
a few parameters, such as, the mean degree and a degree correlation
coefficient. In addition, in the unstable regime we characterize the nature of
the destabilizing mode, which also exhibits universal features. These results
follow from an exact theory for the leading eigenvalue of infinitely large
graphs that are locally tree-like and oriented, as well as, for the right and
left eigenvectors associated with the leading eigenvalue. We corroborate
analytical results for infinitely large graphs with numerical experiments on
random graphs of finite size. We discuss how the presented theory can be
extended to graphs with diagonal disorder and to graphs that contain
nondirected links. Finally, we discuss the influence of small cycles and how
they can destabilize large dynamical systems when they induce strong enough
feedback loops.
