Moments of Two-Variable Functions and the Uniqueness of Graph Limits

Borgs, Christian; Chayes, Jennifer; Lovász, László

doi:10.1007/s00039-010-0044-0

Cited by 116 publications

(214 citation statements)

References 10 publications

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“…However, by now there are analytic proofs: Janson and Diaconis [13] showed that it also follows from results of Hoover and Kallenberg on exchangeable arrays. A direct (and far from simple) proof has recently been given by Borgs, Chayes and Lovász [8].…”

Section: Consequences For Branching Processesmentioning

confidence: 99%

The Cut Metric, Random Graphs, and Branching Processes

2010

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In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in [4], as well as related results of Bollobás, Borgs, Chayes and Riordan [3], all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering [5].

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Section: Consequences For Branching Processesmentioning

confidence: 99%

The Cut Metric, Random Graphs, and Branching Processes

2010

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“…Lovász and Szegedy [13] showed that every convergent sequence of graphs has a limiting graphon. This graphon need not be unique, but it is unique up to a certain type of measure preserving transformation [1], which in some sense corresponds to permutations of the vertices of a graph; two graphons equivalent in this sense are called weakly isomorphic. See [1,10] for more details.…”

Section: Resultsmentioning

confidence: 99%

“…, z m k and add the polynomial p − 1 to P , getting a set P ′ such that the cut norm between W P and W P ′ is very small (for the same values of z i ). We eventually set Z to be the intersection of Z k × [0,1] N over all k ∈ N. The existence of the subsets Z k follows from an application of the Implicit Function Theorem. During the proof, we need to prevent the intersection of Z k × [0, 1] N from becoming degenerate (to guarantee the existence of the bijective function f in Theorem 3).…”

Section: Theorem 3 There Exist a Family Of Graphonsmentioning

confidence: 99%

Extremal graph theory and finite forcibility

Grzesik

Kráľ

Lovász

2017

Electronic Notes in Discrete Mathematics

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We study the uniqueness of optimal solutions to extremal graph theory problems. Our main result is a counterexample to the following conjecture of Lovász, which is often referred to as saying that "every extremal graph theory problem has a finitely forcible optimum": every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints such that the resulting set is satisfied by an asymptotically unique graph.

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“…Austin [1], Bollobás and Riordan [3], Borgs, Chayes and Lovász [4], Lovász [16], Diaconis and Janson [10], Janson [13]. We recall only a few definitions; these will help to fix our notation.…”

Section: Graph Limits and Kernelsmentioning

confidence: 99%

“…Since every kernel is equivalent to some kernel on [0, 1], every graph limit may be represented by a kernel W on [0, 1], equipped with Lebesgue measure λ, but even then W is not unique. Detailed results are in Borgs, Chayes and Lovász [4], Bollobás and Riordan [3] and Janson [13].…”

Section: Graph Limitsmentioning

confidence: 99%

An example of graph limits of growing sequences of random graphs

Janson

Severini

2013

Journal of Combinatorics

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In this paper, we consider a class of growing random graphs obtained by creating vertices sequentially one by one. At each step, we uniformly choose the neighbors of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution, depending on the number of existing vertices. Examples from this class turn out to be the Erdős-Rényi random graph, a natural random threshold graph, etc. By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph. Moreover, for a subclass of models, the growing graph on any given n vertices has the same distribution as the random graph with n vertices that the kernel defines. The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration.

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Moments of Two-Variable Functions and the Uniqueness of Graph Limits

Abstract: For a symmetric bounded measurable function W on [0, 1] 2 and a simple graph F , the homomorphism density

Cited by 116 publications

References 10 publications

The Cut Metric, Random Graphs, and Branching Processes

The Cut Metric, Random Graphs, and Branching Processes

Extremal graph theory and finite forcibility

An example of graph limits of growing sequences of random graphs

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