Delocalization and Limiting Spectral Distribution of Erd\H{o}s-R\'{e}nyi Graphs with Constant Expected Degree

TL;DR

This paper studies the spectral properties of Erdős-Rényi graphs with large constant expected degree, showing that after appropriate weighting and scaling, their spectral distribution converges to a semicircle law and most eigenvectors delocalize.

Contribution

It demonstrates the convergence of the spectral distribution to a semicircle law and establishes eigenvector delocalization for Erdős-Rényi graphs with large expected degree.

Findings

Spectral distribution converges to semicircle law as degree increases.

Most eigenvectors become delocalized in the infinity norm for large degrees.

Edge weighting by 1/√λ is key to the spectral convergence.

Abstract

We consider Erd\H{o}s-R\'{e}nyi graphs $G(n,p_n)$ with large constant expected degree $\lambda$ and $p_n=\lambda/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($\lambda$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\to\infty$. We show that if one weights the edges by $1/\sqrt{\lambda}$ and sends $\lambda\to\infty$, then the support mostly vanishes and in fact, the limiting spectral distributions converge weakly to a semicircle distribution. We also find that for large $\lambda$, there is an orthonormal eigenvector basis of $G(n,p_n)$ such that most of the vectors delocalize with respect to the infinity norm, as $n\to\infty$. Our delocalization result provides a variant on a result of Tran, Vu and Wang (2013).