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This paper presents a method for estimating graphons (graph limits) from sparse graphs by transforming them into their line graphs. The researchers demonstrate that certain sparse graphs with a "square-degree property," such as star graphs and superlinear preferential attachment graphs, produce dense line graphs that enable meaningful graphon estimation, while the original sparse graphs would converge to uninformative zero graphons. This transformation allows researchers to distinguish between different sparse graph structures that would otherwise be indistinguishable using conventional graphon analysis.
Why it matters
This work provides a new analytical tool for studying sparse networks, which are common in real-world applications like social networks, biological networks, and communication systems. The method could improve our ability to characterize and compare large-scale sparse network structures that have previously been difficult to analyze using graph limit theory.
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⚠️ Preprint – Noch nicht peer-reviewed
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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.
Source: Graphons of Line Graphs