The Spectral Distribution of Random Mixed Graphs

Yue Guan,Jianxi Liu,Jinxun Wang,Minfeng Chen,Bo Cheng,Meili Liang,Li Zeng,Chao Yang

doi:10.3390/axioms11030126

Abstract

In this work, we propose a random mixed graph model Gn(p(n),q(n)) that incorporates both the classical Erdős-Rényi’s random graph model and the random oriented graph model. We show that the empirical spectral distribution of Gn(p(n),q(n)) converges to the standard semicircle law under some mild condition, and the Monte Carlo simulation highly agrees with our result.

Highlights

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The Hermitian adjacency matrix is defined in such a way that the digons may be thought of as undirected edges. From this point of view, digraphs are equivalent to the class of mixed graphs we consider here
We find nσ2 increases simultaneously with p, q and the empirical spectral distribution (ESD) is getting more and more close to the semicircle distribution

Summary

Introduction

The Hermitian adjacency matrix is defined in such a way that the digons (i.e., a pair of arcs with the same end vertices but in opposite directions) may be thought of as undirected edges From this point of view, digraphs are equivalent to the class of mixed graphs we consider here. If we set p = 0, our model Gn (0, q/2) is essentially the random oriented graph model since the skew-adjacency matrix S( G ) = iH ( G ) for an oriented graph G N=1 be a sequence of Hermitian adjacency matrices of random oriented graphs { Gn (0, q/2)}∞.

Proof of Fact 1

Proof of Fact 2

Proof of Fact 3

Proof of Theorem 1

Conclusions