Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs

TL;DR

This paper establishes a central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs, providing insights into their spectral properties under certain degree assumptions.

Contribution

It introduces a CLT for the principal eigenvalue and eigenvector of Chung-Lu graphs, extending spectral analysis in inhomogeneous random graph models.

Findings

CLT for principal eigenvalue and eigenvector established

Results depend on degree assumptions ensuring connectivity and sparsity

Provides theoretical foundation for spectral analysis of inhomogeneous graphs

Abstract

A Chung-Lu random graph is an inhomogeneous Erd\H{o}s-R\'enyi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.