The Kato--Temple inequality and eigenvalue concentration with applications to graph inference

TL;DR

This paper adapts the Kato-Temple inequality to provide high-probability bounds on eigenvalue perturbations in random graphs, aiding statistical inference tasks like hypothesis testing and change-point detection.

Contribution

It introduces a novel adaptation of the Kato-Temple inequality for eigenvalue bounds in random graphs, including cases with eigenvalue multiplicity and general noise.

Findings

Explicit bounds for eigenvalue deviations with high probability

Extension to singular value perturbations under general noise

Applications to hypothesis testing and change-point detection in graphs

Abstract

We present an adaptation of the Kato--Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph's adjacency matrix and the corresponding eigenvalues of the model's edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix noise.