### Abstract

In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

Original language | English |
---|---|

Pages (from-to) | 214-225 |

Number of pages | 12 |

Journal | Journal of Graph Theory |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 2012 |

### Fingerprint

### Keywords

- graph limit
- graphon
- random graph model
- weak Positivstellensatz

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*70*(2), 214-225. https://doi.org/10.1002/jgt.20611

**Random graphons and a weak Positivstellensatz for graphs.** / Lovász, L.; Szegedy, Balázs.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 70, no. 2, pp. 214-225. https://doi.org/10.1002/jgt.20611

}

TY - JOUR

T1 - Random graphons and a weak Positivstellensatz for graphs

AU - Lovász, L.

AU - Szegedy, Balázs

PY - 2012/5

Y1 - 2012/5

N2 - In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

AB - In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

KW - graph limit

KW - graphon

KW - random graph model

KW - weak Positivstellensatz

UR - http://www.scopus.com/inward/record.url?scp=84860692173&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860692173&partnerID=8YFLogxK

U2 - 10.1002/jgt.20611

DO - 10.1002/jgt.20611

M3 - Article

AN - SCOPUS:84860692173

VL - 70

SP - 214

EP - 225

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 2

ER -