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We studied the single-particle Anderson localization problem fornon-Hermitian systems on directed graphs. Random regular graph and variousundirected standard random graph models were modified by controllingreciprocity and hopping asymmetry parameters. We found the emergence of left,biorthogonal and right localized states depending on both parameters and graphstructure properties such as node degree $d$. For directed random graphs, theoccurrence of biorthogonal localization near exceptional points is describedanalytically and numerically. The clustering of localized states near thecenter of the spectrum and the corresponding mobility edge for left and rightstates are shown numerically. Structural features responsible for localization,such as topologically invariant nodes or drains and sources, were alsodescribed. Considering the diagonal disorder, we observed the disappearance oflocalization dependence on reciprocity around $W \sim 20$ for a random regulargraph $d=4$. With a small diagonal disorder, the average biorthogonal fractaldimension drastically reduces. Around $W \sim 5$ localization scars occurwithin the spectrum, alternating as vertical bands of clustering of left andright localized states.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry