A graph limit approach to seriation
Date
2023
Authors
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Publisher
University of Delaware
Abstract
The study of graph limit theory began in earnest in 2006 with the publication of the paper Limits of dense graph sequences by Lovász and Szegedy. Through the lens of homomorphism densities—the probability that a random map from a fixed graph F to a graph G is a homomorphism—this paper introduced new tools to the mathematical world that allowed sufficiently large networks to be viewed as random samples from a suitably chosen symmetric function w : [0, 1]2 → [0, 1] called a graphon. Furthermore, Lovász and Szegedy showed that this method of convergence is equivalent to convergence in a specific norm known as the cut norm, where graphs are embedded into [0, 1]2 as step function in the form of their adjacency matrices and the distance is calculated between these step functions. This allows for graph theoretical questions to be translated into questions about step functions, where more standard tools of analysis can be employed. Furthermore, in the reverse direction, one can study how analytical properties of the function can affect the combinatorial properties of its sampled graphs. ☐ We address this question in the context of geometric graphs, whose edge structure is derived from a linear embedding in R. This forces their adjacency matrices to increase toward the main diagonal, also known as the Robinson property, and one that easily translates to graphons. Given a convergent sequence of graphs that become increasingly close to geometric graphs—in the sense of the cut norm—can one claim the limiting object must also be Robinson? This question was solved completely in the affirmative for dense graph sequences, which are sequences of graphs Gn with positive edge density |E(Gn)|/n2 . We therefore focus on graph sequences that are not dense, e.g. whose edge density tends to 0, which corresponds to studying graphons that are unbounded but have finite p-norm. Such graphs are often referred to as being sparse. ☐ Specifically, we introduce and investigate a graph parameter Λ that measures by how much a graphon “fails” to be Robinson, and which is continuous with respect to the cut norm. We also develop a method that constructs Robinson approximations of Lp graphons such that the difference in cut norm between the original graphon and the approximation is dependent on Λ of the original; thus, the closer to being Robinson the original graphon is, the better the approximation.
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Keywords
Cut norm, Graph limit, Graphon, Robinson, Seriation