Convergent sequences of dense graphs I

Subgraph frequencies, metric properties and testing

C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, K. Vesztergombi

Research output: Contribution to journalArticle

198 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1801-1851
Number of pages51
JournalAdvances in Mathematics
Volume219
Issue number6
DOIs
Publication statusPublished - Dec 20 2008

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Convergent Sequence
Subgraph
Metric
Testing
Graph in graph theory
Homomorphisms
Distance Graph
Simple Graph
Partition
Series
Formulation

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, Vol. 219, No. 6, 20.12.2008, p. 1801-1851.

Research output: Contribution to journalArticle

Borgs, C. ; Chayes, J. T. ; Lovász, L. ; Sós, V. T. ; Vesztergombi, K. / Convergent sequences of dense graphs I : Subgraph frequencies, metric properties and testing. In: Advances in Mathematics. 2008 ; Vol. 219, No. 6. pp. 1801-1851.
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