On the Spectra of General Random Mixed Graphs

Abstract

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph $G$ of order $n$ is the $n \times n$ matrix $H(G)=(h_{ij})$, where $h_{ij}=-h_{ji}= \boldsymbol{\mathrm{i}}$ (with $\boldsymbol{\mathrm{i}} =\sqrt{-1})$ if there exists an arc from $v_i$ to $v_j$ (but no arc from $v_j$ to $v_i$), $h_{ij}=h_{ji}=1$ if there exists an edge (and no arcs) between $v_i$ and $v_j$, and $h_{ij}= 0$ otherwise (if $v_i$ and $v_j$ are neither joined by an edge nor by an arc). We study the spectra of the Hermitian adjacency matrix and the normalized Hermitian Laplacian matrix of general random mixed graphs, i.e., in which all arcs are chosen independently with different probabilities (and an edge is regarded as two oppositely oriented arcs joining the same pair of vertices). For our first main result, we derive a new probability inequality and apply it to obtain an upper bound on the eigenvalues of the Hermitian adjacency matrix. Our second main result shows that the eigenvalues of the normalized Hermitian Laplacian matrix can be approximated by the eigenvalues of a closely related weighted expectation matrix, with error bounds depending on the minimum expected degree of the underlying undirected graph.