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

March 16, 2018

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

1 / 19

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

2 / 19

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

2 / 19

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

3 / 19

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

4 / 19

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

4 / 19

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

5 / 19

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 v3

v4

v1

Jakub Sosnovec

v5

v2

Finitely forcible graphons with an almost arbitrary structure

5 / 19

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

Jakub Sosnovec

v3

v4

v1

v5

v2

Finitely forcible graphons with an almost arbitrary structure

5 / 19

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

5 / 19

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

6 / 19

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

6 / 19

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

7 / 19

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

7 / 19

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 .

1 2 3

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 / 19

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 .

1 2 3

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 / 19

Extremal problems Are finitely forcible graphons solutions to extremal problems?

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

8 / 19

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

i=1

Finitely forcible graphons with an almost arbitrary structure

8 / 19

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

Jakub Sosnovec

i=1

Finitely forcible graphons with an almost arbitrary structure

8 / 19

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

8 / 19

Structure of finitely forcible graphons

Do finitely forcible graphons have simple structure?

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

9 / 19

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

9 / 19

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

10 / 19

Counter-example constructions

A

A0

B

B0

B 00

C

C0

D

A

B

C

D

E

F

G

P

Q

R

A

A

B

A0

C D

B

E

B0 F

B 00

G P

C Q

C0 D

R

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

11 / 19

Universal construction The ultimate construction: Theorem (Cooper, Kr´al’, Martins, 2017): Every graphon is a subgraphon of a finitely forcible graphon. A

B

C

D

E

F

G

P

Q

R

A B C D E F G

WF

P

Q

R

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

12 / 19

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

13 / 19

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

13 / 19

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 ε.

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

13 / 19

Universal construction with 1 − ε 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

14 / 19

Ingredients of the proof Partitioned graphons parts P1 , . . . , PM with the same degree, i.e., for every x ∈ Pi , Z degW (x) = W (x, y ) dy = di . [0,1]

Set-decorated constraints method that allows “local forcing” of tiles Pi × Pj of a partitioned graphon Ai

Aj

Bi

Bj

C

D

= C

D

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

15 / 19

Forcing families

Universal forcing family?

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

16 / 19

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

16 / 19

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

16 / 19

More proof details Evolution of constraints: 1

simple density expressions

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

17 / 19

More proof details Evolution of constraints: 1

simple density expressions

2

combinations of density expressions

+2

Jakub Sosnovec

·

Finitely forcible graphons with an almost arbitrary structure

17 / 19

More proof details Evolution of constraints: 1

simple density expressions

2

combinations of density expressions

3

rooted density expressions

·

+2

+2

·

constraint satisfied if true for almost every choice of roots being a partitioned graphon can be forced

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

17 / 19

More proof details Evolution of constraints: 1

simple density expressions

2

combinations of density expressions

3

rooted density expressions

·

+2

·

+2

constraint satisfied if true for almost every choice of roots being a partitioned graphon can be forced B

B

4

decorated density expressions

B

A

A

B

B

A

·

+2 A

vertices are conditioned on belonging to the decorating parts having a finitely forcible graphon on a diagonal tile can be forced

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

17 / 19

More proof details Evolution of constraints: 1

simple density expressions

2

combinations of density expressions

3

rooted density expressions

·

+2

·

+2

constraint satisfied if true for almost every choice of roots being a partitioned graphon can be forced B

B

4

decorated density expressions

B

A

A

B

B

A

·

+2 A

vertices are conditioned on belonging to the decorating parts having a finitely forcible graphon on a diagonal tile can be forced A

B 5

set-decorated density expressions

Jakub Sosnovec

B

A

A

B

B

A

·

+2 A

Finitely forcible graphons with an almost arbitrary structure

17 / 19

Technical difficulties with degree balancing part G1 serves to balance degrees 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

18 / 19

Technical difficulties with degree balancing part G1 serves to balance degrees 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

CKM graphon: 10 parts our graphon: Ω(1/ε) parts Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

18 / 19

Technical difficulties with degree distinguishing part G2 serves to distinguish degrees 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

19 / 19

Technical difficulties with degree distinguishing part G2 serves to distinguish degrees 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

We need to ensure that different parts have different degrees.

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

19 / 19

Technical difficulties with degree distinguishing part G2 serves to distinguish degrees 1−ε

A

ε/4

ε/2

ε/4

W

B C D E F

G1

G2

We need to ensure that different parts have different degrees. Values of W0 on the part G2 are irrational number independent over the rationals. Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

19 / 19

Thank you for your attention.

Jakub Sosnovec

Finitely forcible graphons with an almost arbitrary structure

20 / 19