Measures on the square as sparse graph limits

TL;DR

This paper introduces a new metric for finite graphs based on their bounded dimensional factors, leading to a representation of sparse graph limits via symmetric measures on [0,1]^2, thus extending dense graph limit theory.

Contribution

It proposes a novel metric on finite graphs and generalizes the dense graph limit theory to sparse graph sequences using measure representations.

Findings

Limits of convergent sparse graphs can be represented by symmetric measures.

The new metric captures similarity based on bounded dimensional factors.

Extends the existing dense graph limit framework to sparse graphs.

Abstract

We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on $[0,1]^2$. This leads to a generalization of dense graph limit theory to sparse graph sequences.