Graphons of Line Graphs

TMLR Paper5359 Authors

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 \textit{line graphs}.
We show that graphs satisfying a particular property are sparse, but give rise to dense line graphs.
This property, the \textit{square-degree property}, enables us to apply 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 Erdős–Rényi graphs make the line graphs sparse, resulting in the zero graphon.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=TlDUbCGLuX

Changes Since Last Submission: Formatting update according to TMLR template.

Assigned Action Editor: ~Diana_Cai1

Submission Number: 5359