Linear embeddings of graphs and graph limits

Abstract

Abstract. Consider a random graph process where vertices are chosen from the interval [0,1], and edges are chosen independently at random, but so that, for a given vertex x, the probability that there is an edge to a vertex y decreases as the distance between x and y increases. We call this a random graph with a linear embedding.We define a new graph parameter Γ * , which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph G, Γ * (G) = 0 if and only if G is a unit interval graph, and thus a deterministic example of a graph with a linear embedding.We show that the behaviour of Γ * is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on [0, 1] 2 . We define an operator Γ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence {Gn} converges to a function w, then {Γ * (Gn)} converges as well. Moreover, there exists a function w * arbitrarily close to w under the box distance, so that limn→∞ Γ * (Gn) is arbitrarily close to Γ(w * ).

Uniform linear embeddings of graphons

“…Assume without loss of generality that π(0) < π(r * 1 (0)), and π(0) = 0. From (5) and (6), we obtain important properties for a uniform linear embedding π. These are listed in Corollaries 4.4 and Proposition 4.6 below.…”

“…The notion of diagonally increasing functions, and our interpretation of spatial random graphs, were first given in previous work, see [5]. In [5], a graph parameter Γ is given which aims to measure the similarity of a graph to an instance of a one-dimensional spatial random graph model. However, the parameter Γ fails to distinguish uniform spatial random graph models from the ones which are intrinsically nonuniform.…”

Reconstruction of line-embeddings of graphons

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Consider a random graph process with n vertices corresponding to points v i i.i.d.∼ Unif[0, 1] embedded randomly in the interval, and where edges are inserted between v i , v j independently with probability given by the graphon w(v i , v j ) ∈ [0, 1]. Following [11], we call a graphon w diagonally increasing if, for each x, w(x, y) decreases as y moves away from x. We call a permutation σ ∈ Sn an ordering of these vertices if v σ(i) < v σ(j) for all i < j, and ask: how can we accurately estimate σ from an observed graph?

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An Optimization Parameter for Seriation of Noisy Data

“…This fact becomes apparent when we analyze a growing sequence of graphs which are convergent in the sense of Lovász-Szegedy [18]. Indeed, this article (and the choice of notation for the parameter Γ 1 ) was motivated by our previous work [6], where we introduce a parameter Γ which characterizes Robinson graphons. Graphons are symmetric functions on [0, 1] 2 with values in [0, 1], which can be thought of as the "blueprint" of a random graph whose vertices are randomly sampled from the interval [0, 1].…”

“…Our main result in [6] is that Γ becomes a continuous parameter, when the space of graphons is equipped with the box-norm. Therefore Γ provides us with a parameter to measure Robinson resemblance, which can be efficiently approximated.…”