Graphons of Line Graphs

TL;DR

This paper introduces a method using line graphs to estimate graphons from sparse graphs, enabling analysis of certain sparse graph classes by transforming them into dense line graphs.

Contribution

The paper presents a novel approach that leverages line graphs to infer graph limits of sparse graphs, expanding the applicability of dense graph limit results.

Findings

Star graphs with the square-degree property produce dense line graphs and non-zero graphons.

Superlinear preferential attachment graphs yield dense line graphs almost surely.

Dense graphs like Erdős-Rényi produce sparse line graphs, resulting in zero graphons.

Abstract

We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves mapping the original graphs to their line graphs. We show that graphs satisfying a particular property, which we call the square-degree property are sparse, but give rise to dense line graphs. This enables the use of results on graph limits of dense graphs to derive convergence. In particular, star graphs satisfy the square-degree property resulting in dense line graphs and non-zero graphons of line graphs. We demonstrate empirically that we can distinguish different numbers of stars (which are sparse) by the graphons of their corresponding line graphs. Whereas in the original graphs, the different number of stars all converge to the zero graphon due to sparsity. Similarly, superlinear preferential attachment graphs give rise to dense line graphs almost surely. In contrast, dense graphs, including Erdos-Renyi graphs make the line graphs sparse, resulting in the zero graphon.