Moments of two-variable functions and the uniqueness of graph limits

Christian Borgs, Jennifer Chayes, L. Lovász

Research output: Contribution to journalArticle

60 Citations (Scopus)

Abstract

For a symmetric bounded measurable function W on [0, 1]2 and a simple graph F, the homomorphism density, can be thought of as a "moment" of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (Gn) of dense graphs is said to be convergent if the probability, t(F, Gn), that a random map from V(F) into V(Gn) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]2. Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.

Original languageEnglish
Pages (from-to)1597-1619
Number of pages23
JournalGeometric and Functional Analysis
Volume19
Issue number6
DOIs
Publication statusPublished - Mar 2010

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Measure-preserving Transformations
Uniqueness
Measurable function
Moment
Simple Graph
Homomorphism
Graph in graph theory
Random Maps
Limiting
Converge
Imply

Keywords

  • Convergent graph sequences
  • Graph homomorphisms
  • Graphons
  • Isomorphisms
  • Lebesguian graphon

ASJC Scopus subject areas

  • Geometry and Topology
  • Analysis

Cite this

Moments of two-variable functions and the uniqueness of graph limits. / Borgs, Christian; Chayes, Jennifer; Lovász, L.

In: Geometric and Functional Analysis, Vol. 19, No. 6, 03.2010, p. 1597-1619.

Research output: Contribution to journalArticle

Borgs, Christian ; Chayes, Jennifer ; Lovász, L. / Moments of two-variable functions and the uniqueness of graph limits. In: Geometric and Functional Analysis. 2010 ; Vol. 19, No. 6. pp. 1597-1619.
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