Singular-value statistics of directed random graphs

TL;DR

This paper investigates the singular-value statistics of directed random graphs' adjacency matrices, revealing how these statistics can characterize graph transitions and differentiate between graph models.

Contribution

It provides a numerical analysis of singular-value statistics for directed random graphs, highlighting their effectiveness in identifying graph structural transitions and distinctions.

Findings

Average ratio of nearest neighbor singular values signals graph connectivity transition.

Minimum singular value distribution distinguishes between Erd"os-Rényi and regular graphs.

Singular-value statistics serve as a tool for graph characterization.

Abstract

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.