Regularity partitions and the topology of graphons

TL;DR

This paper explores the topological properties of graphons, showing that those excluding certain subgraphs have compact metric space representations and polynomial-sized regularity partitions.

Contribution

It introduces a unique metric space representation for graphons and links the exclusion of subgraphs to topological and partition size properties.

Findings

Graphons with excluded sub-bigraph have compact metric space representations.

Such graphons have finite packing dimension.

They admit polynomial-sized regularity partitions.

Abstract

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.