### 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 (G_{n}) of dense graphs is said to be convergent if the probability, t(F, G_{n}), that a random map from V(F) into V(G_{n}) 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 |
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Pages (from-to) | 1597-1619 |

Number of pages | 23 |

Journal | Geometric and Functional Analysis |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | Published - Mar 2010 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology
- Analysis

### Cite this

*Geometric and Functional Analysis*,

*19*(6), 1597-1619. https://doi.org/10.1007/s00039-010-0044-0

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

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 19, no. 6, pp. 1597-1619. https://doi.org/10.1007/s00039-010-0044-0

}

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 -