Graphons of Line Graphs
This provides a method to analyze graph limits for sparse graphs, which is incremental but addresses a specific bottleneck in graph theory.
The paper tackles the problem of estimating graph limits (graphons) from sparse finite graphs by mapping them to line graphs, showing that certain sparse graphs like star graphs and superlinear preferential attachment graphs produce dense line graphs with non-zero graphons, enabling convergence analysis where original sparse graphs fail.
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.