2024/09/03 by Sevvandi Kandanaarachchi, Kandanaarachchi, Sevvandi, Cheng Soon Ong +1
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Theory and Algorithms #Machine Learning (cs.LG) #Machine Learning (stat.ML)
paper · pdf · doi:10.48550/arxiv.2409.01656
openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.