Limits of kernel operators and the spectral regularity lemma

TL;DR

This paper explores the spectral properties of graph limits, introduces a spectral regularity lemma, and examines group actions on graphons, revealing differences in invariance properties on spheres versus circles.

Contribution

It provides a spectral perspective on graph limit theory, including a new spectral regularity lemma and analysis of group actions on graphons.

Findings

Graphon convergence characterized by eigenvalues and eigenspaces

Spectral regularity lemma established

Invariance properties differ between sphere and circle for graphons

Abstract

We study the spectral aspects of the graph limit theory. We give a description of graphon convergence in terms of converegnce of eigenvalues and eigenspaces. Along these lines we prove a spectral version of the strong regularity lemma. Using spectral methods we investigate group actions on graphons. As an application we show that the set of isometry invariant graphons on the sphere is closed in terms of graph convergence however the analogous statement does not hold for the circle. This fact is rooted in the representation theory of the orthogonal group.