Uniform linear embeddings of graphons

Abstract

Let w : [0, 1] 2 → [0, 1] be a symmetric function, and consider the random process G(n, w), where vertices are chosen from [0, 1] uniformly at random, and w governs the edge formation probability. Such a random graph is said to have a linear embedding, if the probability of linking to a particular vertex v decreases with distance. The rate of decrease, in general, depends on the particular vertex v. A linear embedding is called uniform if the probability of a link between two vertices depends only on the distance between them. In this article, we consider the question whether it is possible to "transform" a linear embedding to a uniform one, through replacing the uniform probability space [0, 1] with a suitable probability space on R. We give necessary and sufficient conditions for the existence of a uniform linear embedding for random graphs where w attains only a finite number of values. Our findings show that for a general w the answer is negative in most cases.

Reconstruction of line-embeddings of graphons

“…Before we discuss the weaker assumptions, we give an alternative characterization of {0, 1}-valued graphons. It is not hard to verify (see also the discussion following Definition 2.1 of [12]) that, if w is a {0, 1}-valued graphon, then w is diagonally increasing if and only if there exist two non-decreasing functions : [0, 1] → [0, 1] and r : [0, 1] → [0, 1] which demarkate the regions in [0, 1] 2 where w assumes the value 1. The functions and r will be called the boundaries of w. Precisely, w is of the form…”

“…In practice, we expect that it is often easy to diagnose graphons that don't satisfy Assumption 1.8 as follows. Let F (2) be as in step (12) of Algorithm 7 on the first time that it is called by Algorithm 6; if Assumption 1.8 is satisfied, then with extreme probability the maximum size of the set on which disagreement occurs max i,j∈V2\V1 :…”

Uniform Embeddings for Robinson Similarity Matrices

“…Graphons can be seen as generalizations of matrices. The results from [3] do not apply in the context of this paper. Namely, a matrix can be represented as a graphon, but the "boundaries" delineating the regions of [0, 1] 2 where the graphon takes a certain value 1 ≤ t ≤ k in this case are piecewise constant functions.…”

“…In [3], the problem of finding a uniform embedding was studied for diagonally increasing graphons. A graphon is a symmetric function w :…”

An Optimization Parameter for Seriation of Noisy Data

“…When applied to real-life networks, the parameter Γ 1 can be used to measure how closely the matrix conforms to a linear model. In previous work, the authors investigated this question in the context of graph limits [6,7]. In a linear graph model the vertices of the graph are placed on a line, and the links are formed stochastically so that vertices that are closer together are more likely to connect.…”