  Although the spectral properties of random graphs have been a long-standing
focus of network theory, the properties of right eigenvectors of directed
graphs have so far eluded an exact analytic treatment. We present a general
theory for the statistics of the right eigenvector components in directed
random graphs with a prescribed degree distribution and with randomly weighted
links. We obtain exact analytic expressions for the inverse participation ratio
and show that right eigenvectors of directed random graphs with a small average
degree are localized. Remarkably, if the fourth moment of the degree
distribution is finite, then the critical mean degree of the localization
transition is independent of the degree fluctuations, which is different from
localization in undirected graphs that is governed by degree fluctuations. We
also show that in the high connectivity limit the distribution of the right
eigenvector components is solely determined by the degree fluctuations. For
delocalized eigenvectors, we recover the universal results from standard random
matrix theory that are independent of the degree distribution, while for
localized eigenvectors the eigenvector distribution depends on the degree
distribution.
