Before revision: We develop a mathematical theory for the linear stability of stationary states  in large dynamical systems modelled by a set of randomly coupled differential equations on a locally tree-like network. Our approach provides analytical expressions for the leading eigenvalue of random matrices that describe the interactions between the degrees of freedom; the sign of the leading eigenvalue characterizes the system stability. We illustrate this approach on oriented random graphs with a prescribed degree distribution and find that the leading eigenvalue is universal in the sense that it only depends on a few ensemble parameters, including the mean degree and a degree correlation coefficient.
After revision: We develop a mathematical theory for the linear stability of stationary states 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 only depends on a few ensemble parameters, including the mean degree and a degree correlation coefficient.