Singular-value statistics of directed random graphs

Singular-value statistics (SVS) has been recently presented as a random matrix theory tool able to properly characterize non-Hermitian random matrix ensembles [PRX Quantum {\bf 4}, 040312 (2023)]. Here, we perform a numerical study of the SVS of the non-Hermitian adjacency matrices $\mathbf{A}$ of directed random graphs, where $\mathbf{A}$ are members of diluted real Ginibre ensembles. We consider two models of directed random graphs: Erd\"os-R\'enyi graphs and random regular graphs. Specifically, we focus on the ratio $r$ between nearest neighbor singular values and the minimum singular value $\lambda_{min}$. We show that $\langle r \rangle$ (where $\langle \cdot \rangle$ represents ensemble average) can effectively characterize the transition between mostly isolated vertices to almost complete graphs, while the probability density function of $\lambda_{min}$ can clearly distinguish between different graph models.