The Spectral Distribution of Random Graphs with Given Degree Sequences

TL;DR

This paper proves that the spectral distribution of normalized Laplacians in random graphs with specified degree sequences converges to the semicircle law, extending classical results from regular graphs to more general degree sequences.

Contribution

It extends the spectral convergence results from regular graphs to graphs with general degree sequences under mild assumptions.

Findings

Spectral distribution converges to the semicircle law as n→∞.

Generalized conditions are equivalent to convergence to the semicircle distribution.

Numerical simulations support the theoretical results.

Abstract

In this article, we study random graphs with a given degree sequence $d_1, d_2, \cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian matrix of such random graph converges in distribution to the semicircle distribution as the number of vertices $n\rightarrow \infty$. This extends work by McKay (1981) and Tran, Vu and Wang (2013) which studied random regular graphs ($d_1=d_2=\cdots=d_n=d$). Furthermore, we extend the assumption to show that a slightly more general condition is equivalent to the weak convergence to semicircle distribution. The equivalence is also illustrated by numerical simulations.