Random graphons and a weak Positivstellensatz for graphs

L. Lovász, Balázs Szegedy

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Pages (from-to)214-225
Number of pages12
JournalJournal of Graph Theory
Volume70
Issue number2
DOIs
Publication statusPublished - May 2012

Fingerprint

Graph in graph theory
Consistency Conditions
Valid Inequalities
Formal Proof
Sum of squares
Graph Model
Random Graphs
Probability Measure
Countable
Linear Inequalities
Subgraph
Multiplicative
Permutation
Analogue
Algebra
Vertex of a graph

Keywords

  • graph limit
  • graphon
  • random graph model
  • weak Positivstellensatz

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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

In: Journal of Graph Theory, Vol. 70, No. 2, 05.2012, p. 214-225.

Research output: Contribution to journalArticle

Lovász, L. ; Szegedy, Balázs. / Random graphons and a weak Positivstellensatz for graphs. In: Journal of Graph Theory. 2012 ; Vol. 70, No. 2. pp. 214-225.
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