Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erd\H{o}s-R\'enyi Random Graphs

TL;DR

This paper analyzes the spectral properties of adjacency and Laplacian matrices of inhomogeneous Erdős-Rényi random graphs in the sparse regime, establishing convergence of empirical spectral distributions to deterministic limits and providing explicit descriptions for certain cases.

Contribution

It introduces a framework for understanding the spectral distribution of inhomogeneous sparse random graphs and characterizes the limits explicitly for specific kernel functions.

Findings

Empirical spectral distributions converge to deterministic limits.

Explicit characterization for the case where f(x,y) = r(x)r(y).

Applications to real-world network models.

Abstract

Inhomogeneous Erd\H{o}s-R\'enyi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N\,f(\frac{i}{N},\frac{j}{N})$, $1 \leq i \neq j \leq N$, independently of other edges, where $f\colon\,[0,1] \times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $\Delta_N$ associated with $\mathbb G_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, it is shown that the empirical spectral distributions of $A_N$ and $\Delta_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\,[0,1] \to [0,\infty)$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.