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.

翻译：奇异值统计（SVS）近期被提出作为一种能够恰当表征非厄米随机矩阵系综的随机矩阵理论工具 [PRX Quantum {\bf 4}, 040312 (2023)]。本文针对有向随机图的非厄米邻接矩阵 $\mathbf{A}$（属于稀释实Ginibre系综）的SVS进行数值研究。我们考虑两种有向随机图模型：Erd\"os-R\'enyi图与随机正则图。具体而言，我们聚焦于最近邻奇异值比值 $r$ 与最小奇异值 $\lambda_{min}$。研究表明，$\langle r \rangle$（其中 $\langle \cdot \rangle$ 表示系综平均）可有效表征从绝大多数孤立顶点到几乎完全图的转变过程，而 $\lambda_{min}$ 的概率密度函数能够清晰区分不同图模型。