# 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 language English 1597-1619 23 Geometric and Functional Analysis 19 6 https://doi.org/10.1007/s00039-010-0044-0 Published - Mar 2010

### Fingerprint

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.
@article{a99e1d3cc0524ee29a50ca6f3f161bd8,
title = "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, 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.",
keywords = "Convergent graph sequences, Graph homomorphisms, Graphons, Isomorphisms, Lebesguian graphon",
author = "Christian Borgs and Jennifer Chayes and L. Lov{\'a}sz",
year = "2010",
month = "3",
doi = "10.1007/s00039-010-0044-0",
language = "English",
volume = "19",
pages = "1597--1619",
journal = "Geometric and Functional Analysis",
issn = "1016-443X",
publisher = "Birkhauser Verlag Basel",
number = "6",

}

TY - JOUR

T1 - Moments of two-variable functions and the uniqueness of graph limits

AU - Borgs, Christian

AU - Chayes, Jennifer

AU - Lovász, L.

PY - 2010/3

Y1 - 2010/3

N2 - 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.

AB - 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.

KW - Convergent graph sequences

KW - Graph homomorphisms

KW - Graphons

KW - Isomorphisms

KW - Lebesguian graphon

UR - http://www.scopus.com/inward/record.url?scp=77952958250&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952958250&partnerID=8YFLogxK

U2 - 10.1007/s00039-010-0044-0

DO - 10.1007/s00039-010-0044-0

M3 - Article

AN - SCOPUS:77952958250

VL - 19

SP - 1597

EP - 1619

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 6

ER -