Regularity Partitions and The Topology of Graphons

Lovász, László; Szegedy, Balázs

doi:10.1007/978-3-642-14444-8_12

Cited by 79 publications

(102 citation statements)

References 15 publications

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“…One is to show that our regularity lemma is well behaved with respect to counting and the second goal is to show that function sequences in which the density of every fixed configuration converges have a nice limit object which is a measurable function on a nilspace. This fits well into the recently developed graph and hypergraph limit theories [17], [2], [18], [19], [3].…”

Section: Then For Functions On Groups In a Theorem 2 Holds With A-nilsupporting

confidence: 84%

“…Topology and geometry does not play a direct role in stating the regularity lemma for graphs. (Note that a connection of Szemerédi's regularity lemma to topology was highlighted in [19].) However in the abelian group case, even if we just regularize functions on finite abelian groups, compact geometric structures come up naturally as target spaces of the morphism φ.…”

Section: To Summarize the Results In This Paper We Start With The Defmentioning

confidence: 99%

“…This means that the use of ordinary Fourier analysis is restricted to functions that are measurable in F 1 . We will need higher order generalizations ofÂ to define the analogy of (18) and (19) for functions that are measurable in F k . Note that the definition implies that every element ofÂ k is a shift invariant rank one module of order k. We will see later thatÂ k could be equivalently defined as the set of shift invariant rank one modules of order k. To justify the name "higher order dual group" we need to give a group structure to it.…”

Section: Remark 31 Let X Be An Affine Version Of a (See Chapter 23)mentioning

confidence: 99%

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Higher Order Fourier Analysis

Tao

2012

Graduate Studies in Mathematics

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We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms.We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact k-step nilspaces. As a byproduct of our results we obtain a new type of limit theory for functions on abelian groups in the spirit of the socalled graph limit theory. Our proofs are based on an exact (non-approximative) version of higher order Fourier analysis which appears on ultra product groups.

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Section: Then For Functions On Groups In a Theorem 2 Holds With A-nilsupporting

confidence: 84%

Section: To Summarize the Results In This Paper We Start With The Defmentioning

confidence: 99%

Section: Remark 31 Let X Be An Affine Version Of a (See Chapter 23)mentioning

confidence: 99%

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Higher Order Fourier Analysis

Tao

2012

Graduate Studies in Mathematics

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“…It is easy to see that R(W ) ∩ T (W ) is dense in T (W ) ( This topological space was introduced and studied in [18], where it was proved that if t(F , W ) = 0 for some signed bipartite graph F , then T (W ) is finite dimensional and compact. The following conjectures would lead to the same conclusion from a different assumption.…”

Section: Weak Homogeneitymentioning

confidence: 97%

Finitely forcible graphons

Lovász

Szegedy

2011

Journal of Combinatorial Theory, Series B

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We investigate families of graphs and graphons (graph limits) that are determined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turán, Erdős-Simonovits and Chung-Graham-Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractal-like) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are "rare", and exhibit simple and explicit non-forcible graphons.

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“…(See also the simple arrays of [34] and 0-1 valued graphons in [47].) When W is random-free, the W -random graph process amounts, in the language of [52], to 'randomization in vertices' but not 'randomization in edges'.…”

Section: Applications and Further Observationsmentioning

confidence: 99%

Invariant Measures Concentrated on Countable Structures

Freer

Patel

2016

Forum of Mathematics, Sigma

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Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

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Regularity Partitions and The Topology of Graphons

Abstract: We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon

Cited by 79 publications

References 15 publications

Higher Order Fourier Analysis

Higher Order Fourier Analysis

Finitely forcible graphons

Invariant Measures Concentrated on Countable Structures

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