Fractional Isomorphism of Graphons

TL;DR

This paper extends the concept of fractional isomorphism from finite graphs to graphons, providing a comprehensive theoretical framework involving homomorphism densities, Markov operators, and measure distributions.

Contribution

It introduces a generalized theory of fractional isomorphism for graphons, connecting it with homomorphism densities, measure distributions, and operator isomorphisms.

Findings

Characterizes fractional isomorphism of graphons via homomorphism densities of finite trees.

Establishes equivalences using distributions on iterated degree measures and Markov operators.

Provides a framework for understanding graphon isomorphism through quotient structures.

Abstract

We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is characterized in terms of distributions on iterated degree measures, Markov operators, weak isomorphism of a conditional expectation with respect to invariant sub-$\sigma$-algebras and isomorphism of certain quotients of given graphons.