Before revision:  In addition, we also characterize the unstable mode of the system of interest by deriving analytical expressions for the statistics of the components of the right and left eigenvectors associated with the leading eigenvalue.
After revision:  In addition, we also characterize the unstable mode of the system of interest by deriving analytical expressions for the statistics of 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.