The Large Deviation Principle for $W$-random spectral measures

TL;DR

This paper establishes a Large Deviation Principle for spectral measures of $W$-random graph-based integral operators, revealing how eigenvalues and eigenspaces are influenced by large deviations in the underlying graphon.

Contribution

It extends the LDP framework to spectral measures of symmetric Hilbert-Schmidt integral operators derived from $W$-random graphs, with continuous dependence and asymptotic analysis.

Findings

Eigenvalues and eigenspaces are affected by large deviations in the graphon.

Provides asymptotic behavior of eigenvalues for certain random graph sequences.

Demonstrates continuous dependence of spectral measures on graphons.

Abstract

The $W$-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for $W$-random graphs from [9], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by the large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the underlying graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of the integral operators corresponding to certain random graph sequences. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces of the integral operators are affected by large deviations. Potential implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing are indicated.