Finitely forcible graphons with an almost arbitrary structure Jakub Sosnovec joint work with Daniel Kr´al’
L´aszl´ o M. Lov´asz
Jonathan A. Noel
University of Warwick
November 24, 2017
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Overview
Limits of dense graphs: |E | = Ω(|V |2 ) Developed by Borgs, Chayes, Lov´asz, S´ os, Szegedy, Vesztergombi,... Applications in extremal combinatorics, computer science, property testing,...
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Overview
Limits of dense graphs: |E | = Ω(|V |2 ) Developed by Borgs, Chayes, Lov´asz, S´ os, Szegedy, Vesztergombi,... Applications in extremal combinatorics, computer science, property testing,... This talk: 1 2 3 4
Introduction to graphons Finite forcibility Known constructions A new construction and a “lower bound”
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graph convergence
Subgraph density d(H, G ): probability that randomly chosen |H| vertices of G induce a graph isomorphic to H A sequence G1 , G2 , . . . with |Gn | → ∞ is convergent if for every H, the sequence d(H, Gn ) is convergent. Examples: Gn = Kn Gn = Kn,n Gn = G (n, p) sparse graphs (d(K2 , Gn ) → 0)
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons
A graphon is a measurable function W : [0, 1]2 → [0, 1] that satisfies W (x, y ) = W (y , x) for every x, y ∈ [0, 1]
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons
A graphon is a measurable function W : [0, 1]2 → [0, 1] that satisfies W (x, y ) = W (y , x) for every x, y ∈ [0, 1]
infinite weighted graph on [0, 1] continuous version of adjacency matrix probability distribution on graphs
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction
W -random graph of order n: 1 2
x1 , . . . , xn ∈ [0, 1] random points, chosen uniformly and independently edge vi vj present with probability W (xi , xj )
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction
W -random graph of order n: 1 2
x1 , . . . , xn ∈ [0, 1] random points, chosen uniformly and independently edge vi vj present with probability W (xi , xj )
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Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction
W -random graph of order n: 1 2
x1 , . . . , xn ∈ [0, 1] random points, chosen uniformly and independently edge vi vj present with probability W (xi , xj )
x1 x2 x3 x4 x5 x1 x2 x3 x4 x5
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Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction
W -random graph of order n: 1 2
x1 , . . . , xn ∈ [0, 1] random points, chosen uniformly and independently edge vi vj present with probability W (xi , xj )
Subgraph density d(H, W ): probability that a W -random graph of order |H| is isomorphic to H
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction Sequence G1 , G2 , . . . of graphs converges to a graphon W if for every graph H, lim d(H, Gn ) = d(H, W ). n→∞
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Graphons introduction Sequence G1 , G2 , . . . of graphs converges to a graphon W if for every graph H, lim d(H, Gn ) = d(H, W ). n→∞
Examples: (Kn ) converges to W ≡ 1 (G (n, p)) converges to W ≡ p sparse graphs converge to W ≡ 0
Every convergent sequence of graphs has a limit (Lov´asz, Szegedy, 2006). The sequence of W -random graphs of increasing orders converges to W with probability 1.
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Finitely forcible graphons Graphons W and W 0 are weakly isomorphic if d(H, W ) = d(H, W 0 ) for every graph H. weakly isomorphic graphons represent the same limit characterization in terms of measure preserving transformations
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Finitely forcible graphons Graphons W and W 0 are weakly isomorphic if d(H, W ) = d(H, W 0 ) for every graph H. weakly isomorphic graphons represent the same limit characterization in terms of measure preserving transformations
A graphon W is finitely forcible if ∃H1 , . . . , Hm such that ∀W 0 : d(Hi , W 0 ) = d(Hi , W ) ∀i = 1, . . . , m =⇒ W 0 weakly isomorphic to W .
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Finitely forcible graphons Graphons W and W 0 are weakly isomorphic if d(H, W ) = d(H, W 0 ) for every graph H. weakly isomorphic graphons represent the same limit characterization in terms of measure preserving transformations
A graphon W is finitely forcible if ∃H1 , . . . , Hm such that ∀W 0 : d(Hi , W 0 ) = d(Hi , W ) ∀i = 1, . . . , m =⇒ W 0 weakly isomorphic to W .
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E-R: Thomason (1987), Chung, Graham, Wilson (1989) Step graphons: Lov´asz and S´ os (2008) Half graphon: Diaconis, Holmes, Janson (2009)
The collection H1 , . . . , Hm is a forcing family for W . Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
7 / 17
Finitely forcible graphons Graphons W and W 0 are weakly isomorphic if d(H, W ) = d(H, W 0 ) for every graph H. weakly isomorphic graphons represent the same limit characterization in terms of measure preserving transformations
A graphon W is finitely forcible if ∃H1 , . . . , Hm such that ∀W 0 : d(Hi , W 0 ) = d(Hi , W ) ∀i = 1, . . . , m =⇒ W 0 weakly isomorphic to W .
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E-R: Thomason (1987), Chung, Graham, Wilson (1989) Step graphons: Lov´asz and S´ os (2008) Half graphon: Diaconis, Holmes, Janson (2009)
The collection H1 , . . . , Hm is a forcing family for W . Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Extremal problems Are finitely forcible graphons solutions to extremal problems?
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Extremal problems Are finitely forcible graphons solutions to extremal problems? Flag algebras method =⇒ W0 finitely forcible iff ∃H1 , . . . , Hk and reals α1 , . . . , αm such that W0 is the unique minimizer of k X min αi d(Hi , W ). W
Jakub Sosnovec
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Finitely forcible graphons with an almost arbitrary structure
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Extremal problems Are finitely forcible graphons solutions to extremal problems? Flag algebras method =⇒ W0 finitely forcible iff ∃H1 , . . . , Hk and reals α1 , . . . , αm such that W0 is the unique minimizer of k X min αi d(Hi , W ). W
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Conjecture (Lov´asz, Szegedy, 2011): Let α1 , . . . , αk be reals and H1 , . . . , Hk graphs. There exists a finitely forcible graphon that minimizes k X min αi d(Hi , W ). W
Jakub Sosnovec
i=1
Finitely forcible graphons with an almost arbitrary structure
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Extremal problems Are finitely forcible graphons solutions to extremal problems? Flag algebras method =⇒ W0 finitely forcible iff ∃H1 , . . . , Hk and reals α1 , . . . , αm such that W0 is the unique minimizer of k X min αi d(Hi , W ). W
i=1
Conjecture (Lov´asz, Szegedy, 2011): Let α1 , . . . , αk be reals and H1 , . . . , Hk graphs. There exists a finitely forcible graphon that minimizes k X min αi d(Hi , W ). W
i=1
Theorem (Grzesik, Kr´al’, Lov´asz Jr., 2017): This is FALSE. Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Structure of finitely forcible graphons
Do finitely forcible graphons have simple structure?
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Structure of finitely forcible graphons
Do finitely forcible graphons have simple structure? Lov´asz and Szegedy conjectured: Conjecture (LS, 2011): The space of typical vertices of a finitely forcible graphon is compact. Conjecture (LS, 2011): The space of typical vertices of a finitely forcible graphon is finite-dimensional.
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Counter-example constructions
Do finitely forcible graphons have simple structure? The answer is NO. Theorem (Glebov, Kr´al’, Volec, 2013): The space of typical vertices may fail to be locally compact. Theorem (Glebov, Klimoˇsov´a, Kr´al’, 2014): The space of typical vertices may have a part homeomorphic to [0, 1]∞ . Theorem (Cooper, Kaiser, Kr´al’, Noel, 2015): There exists a finitely forcible graphon such that every ε-regular partition has at least −2 −1 2ε / log log ε parts (for infinitely many ε → 0).
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Counter-example constructions
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Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Universal construction The ultimate construction: Theorem (Cooper, Kr´al’, Martins, 2017): Every graphon is a subgraphon of a finitely forcible graphon. A
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Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Universal construction
In fact, the following was proved: Theorem (CKM, 2017): There exists a family of graphs H1 , . . . , Hm such that for every graphon W there exists a finitely forcible graphon W0 such that W0 contains W as a subgraphon on a set of measure 1/13, H1 , . . . , Hm is a forcing family for W0 .
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Universal construction
In fact, the following was proved: Theorem (CKM, 2017): There exists a family of graphs H1 , . . . , Hm such that for every graphon W there exists a finitely forcible graphon W0 such that W0 contains W as a subgraphon on a set of measure 1/13, H1 , . . . , Hm is a forcing family for W0 .
Theorem (Kr´al’, Lov´asz Jr., Noel, S., 2017+): For every ε > 0 and a graphon W , there exists a finitely forcible graphon W0 that contains W as a subgraphon on a set of measure 1 − ε.
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Universal construction
In fact, the following was proved: Theorem (CKM, 2017): There exists a family of graphs H1 , . . . , Hm such that for every graphon W there exists a finitely forcible graphon W0 such that W0 contains W as a subgraphon on a set of measure 1/13, H1 , . . . , Hm is a forcing family for W0 .
Theorem (Kr´al’, Lov´asz Jr., Noel, S., 2017+): For every ε > 0 and a graphon W , there exists a finitely forcible graphon W0 that contains W as a subgraphon on a set of measure 1 − ε. Note: The forcing family of W0 depends on W .
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Universal construction with 1 − ε 1−ε
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Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Ingredients of the proof Partitioned graphons parts P1 , . . . , PM with the same degree, i.e., for every x ∈ Pi , Z degW (x) = W0 (x, y ) dy = di . [0,1]
Set-decorated constraints method that allows “local forcing” of tiles Pi × Pj of a partitioned graphon Ai
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CKM graphon as a black box possible to force finitely forcible graphon on a diagonal tile Pi × Pi
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Forcing families
Universal forcing family?
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Forcing families
Universal forcing family? Let Gn be the set of all graphs of order at most n.
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Forcing families
Universal forcing family? Let Gn be the set of all graphs of order at most n. Theorem (Kr´al’, Lov´asz Jr., Noel, S., 2017+): For every n ∈ N, there exists a real ε = ε(n) > 0 and a graphon Un such that if W0 is a graphon containing Un as a subgraphon on a set of measure at least 1 − ε, then Gn does not force W0 .
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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Thank you for your attention.
Jakub Sosnovec
Finitely forcible graphons with an almost arbitrary structure
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