The automorphism group of a graphon

TL;DR

This paper investigates the automorphism groups of graphons, establishing their compactness after standardization, and characterizes their action on tuples, with applications to graph algebras and node-transitive graphons.

Contribution

It proves the compactness of automorphism groups of graphons after standardization and characterizes their orbits, advancing understanding of graphon symmetries.

Findings

Automorphism group is compact after standardization

Characterization of automorphism group orbits on k-tuples

Applications to finite rank graphons and node-transitive graphons

Abstract

We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on $k$-tuples of points. Among applications we study the graph algebras defined by finite rank graphons and the space of node-transitive graphons.