Localization and Universality of Eigenvectors in Directed Random Graphs

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