### Abstract

We consider sequences of graphs (G_{n}) and define various notions of convergence related to these sequences: "left convergence" defined in terms of the densities of homomorphisms from small graphs into G_{n}; "right convergence" defined in terms of the densities of homomorphisms from G_{n} into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs G_{n}, and for graphs G_{n} with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemerédi partitions, sampling and testing of large graphs.

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

Pages (from-to) | 1801-1851 |

Number of pages | 51 |

Journal | Advances in Mathematics |

Volume | 219 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 20 2008 |

### Fingerprint

### Keywords

- Cut metric
- Free energies
- Graph sequences
- Parameter testing
- Partition functions
- Sampling
- Szemerédi partitions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*219*(6), 1801-1851. https://doi.org/10.1016/j.aim.2008.07.008

**Convergent sequences of dense graphs I : Subgraph frequencies, metric properties and testing.** / Borgs, C.; Chayes, J. T.; Lovász, L.; Sós, V. T.; Vesztergombi, K.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 219, no. 6, pp. 1801-1851. https://doi.org/10.1016/j.aim.2008.07.008

}

TY - JOUR

T1 - Convergent sequences of dense graphs I

T2 - Subgraph frequencies, metric properties and testing

AU - Borgs, C.

AU - Chayes, J. T.

AU - Lovász, L.

AU - Sós, V. T.

AU - Vesztergombi, K.

PY - 2008/12/20

Y1 - 2008/12/20

N2 - We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: "left convergence" defined in terms of the densities of homomorphisms from small graphs into Gn; "right convergence" defined in terms of the densities of homomorphisms from Gn into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs Gn, and for graphs Gn with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemerédi partitions, sampling and testing of large graphs.

AB - We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: "left convergence" defined in terms of the densities of homomorphisms from small graphs into Gn; "right convergence" defined in terms of the densities of homomorphisms from Gn into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs Gn, and for graphs Gn with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemerédi partitions, sampling and testing of large graphs.

KW - Cut metric

KW - Free energies

KW - Graph sequences

KW - Parameter testing

KW - Partition functions

KW - Sampling

KW - Szemerédi partitions

UR - http://www.scopus.com/inward/record.url?scp=53349180210&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53349180210&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2008.07.008

DO - 10.1016/j.aim.2008.07.008

M3 - Article

VL - 219

SP - 1801

EP - 1851

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -