Dense graph limit theory: Extremal graph theory László Lovász Eötvös Loránd University, Budapest…

Dense graph limit theory:Extremal graph theory

László Lovász Eötvös Loránd University, Budapest

May 2012 1

May 2012 2

Recall some math

t(F,G): Probability that random map V(F)V(G) preserves edges

(G1,G2,…) convergent: F t(F,Gn) is convergent

W0 = {W: [0,1]2 [0,1], symmetric, measurable}

( ) ( )[0,1]

( ,( , ) )Î

= ÕòV F

i jij E F

W x x dxt F W

GnW : F: t(F,Gn) t(F,W)

"graphon"

May 2012 3

, [0,1]sup (, ') '( )

XS T S T

W Wd W W

'( , ') inf ( , ')

XX W W

d W WW W

Recall some math

The semimetric space (W0,) is compact.

May 2012 4

Recall some math

For every convergent graph sequence (Gn)

there is a WW0 such that GnW .

W is essentially unique

(up to measure-preserving

transformation).

Conversely, W (Gn) such that GnW .

May 2012

Turán’s Theorem (special case proved by Mantel):G contains no triangles #edgesn2/4

Theorem (Goodman):

3(2 -1) ( )2 3

#edges #triangles n n

c c c o n

Extremal:

5

A sampler of results from extremal graph theory

May 2012

Kruskal-Katona Theorem (very special case):

2 3#edges #triangles

k k

nk

6

Some old and new results from extremal graph theory

May 2012

Semidefiniteness and extremal graph theory Tricky examples

1

10

Kruskal-Katona

Bollobás

1/2 2/3 3/4

Razborov 2006

Mantel-Turán

Goodman

Fisher

Lovász-Simonovits


7

May 2012

Theorem (Erdős):G contains no 4-cycles #edgesn3/2/2

(Extremal: conjugacy graph of finite projective planes)

8


( )4

( , ) ( , ) ( , ) ( , )

( , )³ò

òW x y W y z W z u W u x dxdydzdu

W x y dxdy Cauchy-Schwarztwice

4( , ) ( | , )Wt G t G

May 2012 9

Thomason, Chung-Graham-Wilson

Common properties of random graphs G(n,p) (n):

(1) almost all degrees pn, almost all codegrees

p2n.

(2) for X,YV(G), e(X,Y)= p|X||Y|+o(n2)

(3) for every graph F, t(F,G)p|E(F)|

(4) t( | ,G) p, t( ,G) p4For any sequence of graphs Gn (|V(Gn)|=n ), these properties are equivalent.

Quasirandom graph sequences

Example: Paley graphs

p: prime 1 mod 4


xy E(G) x-y =

May 2012 11


For every graph F, t(F,Gn)p|E(F)| Gnp

For every graph F, t(F,G)p|

E(F)|

t( | ,G) p, t(,G) p4

If t( | ,W)= p, t( , W) =p4, then W p

(equality in Cauchy-Schwarz)

May 2012 12

General questions about extremal graphs

- Which inequalities between subgraph densities are valid?

- Which graphs are extremal?

- Can all valid inequalities be proved using just Cauchy-Schwarz?

- Local vs. global extrema

May 2012 13






May 2012 14

Extremal problems

1

0

( ,: )? ?m

ii it FG G

If valid asymptotically for large G,

then valid for all

May 2012 15

Analogy with polynomials

p(x1,...,xn)0

for all x1,...,xnZ undecidable Matiyasevich

for all x1,...,xnR decidable TarskiÛ p = r1

2 + ...+ rm2 (r1, ...,rm are rational functions)

Artin

May 2012 16

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

May 2012

1

10 1/2 2/3 3/4

17

The main trick in the proof

t( ,G) – 2t( ,G) + t( ,G) = 0 …

May 2012 18

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

…but decidable with an arbitrarily small error.

L-Szegedy

May 2012 19


- Is there always an extremal graph?




May 2012

Write x ≥ 0 if hom(x,G) = xGG ≥ 0 for every graph

G. Turán: -2 + 0³

Kruskal-Katona: - 0³

Computing with graphs

20

Erdős: - 0³

GG

x x G=å formal linear combination of graphs

May 2012

- +-2

= - +-

- +- 2+2 2- = - +- +2 -4 +2

Goodman’s Theorem

Computing with graphs

21

+- 2+- 2 ≥ 0

2- = 2 -4 +2

t( ,G) – 2t( ,G) + t( ,G) ≥ 0

May 2012

Graph parameter: isomorphism-invariant function on finite graphs

k-labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes

1

2

22

Which parameters are homomorphism functions?

May 2012

k=2:

...

...

( )f

M(f, k)

23

Connection matrices

May 2012

Freedman - L - Schrijver

: ( , )k M f k" is positive semidefinite

and has rank ck.

hom(., ) for some weighted graphf H H


24

May 2012

: ( , )k M f k" is positive semidefinite,

f( )=1 and f is multiplicative

(., )f t W

25

L - Szegedy


May 2012

2 21

2 21( ... ) ?...+ ++ = + mn x y yz z

Question: Suppose that x ≥ 0. Does it follow that2 21 ... ?= + + mx y y

Positivstellensatz for graphs?

26

No! Hatami-Norine

If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0.

May 2012

2 21 1 1

Let be a quantum graph. Then 0

0 ,..., ...e e

³

" > $ $ Î - - - <Û

m k m

x x

k y y G x y y

A weak Positivstellensatz

27

L - Szegedy

November 2010

Semidefinite formulation of the Mantel-Turán Theorem

28

G: (large) unknown graph

xF = t(F,G): variables

( , )

___________________

1

2

3

1

positive semide

ma

finite

0

ximize

UÆ=

=

=

FK

K

K

F

xx x

M x

x

k

x

( , )1 21 2

has rank

bounded by

U =

k

F FF Fx x x

M x kc

Can be ignored

Infinitely many variablesmust be cut to finite sizearbitrarily small error?

The optimum of the semidefinite program

minimize

subject to M(x,k) positive semidefinite for all k =1

is 0.

May 2012

Proof of the weak Positivstellensatz (sketch2)

Apply Duality Theorem of semidefinite programming.

29

: ( , ) 0l" ³ Ûå i iG t F G

i iFxlå

1Kx1F K Fx xÈ =

May 2012 30






May 2012 31

Alon-Stav

If P is a hereditary graph property, then

there is a 0≤p≤1 such that G(n,p) is

asymptotically farthest from P among

all n-node graphs.

in “edit distance”

Local optimum: when is it global?

May 2012 32

0

{ : }

\

" Î= Î=

WKPR

W

R

a G

G

G WW G

Want: maximize d1(U,R): UK

K is convex

K is invariant under measure preserving transformations

d1(.,R) is concave on K

d1 is maximized on Kby a constant function

Local optimum: when is it global?

May 2012 33






Finite forcing

Graphon W is finitely forcible:

1 1

1 1,..., , ,...

( , )( , ) ( , )

(

,

, )

:

m

m m

m

t F UF t F U t F W

t F U

F F

a

a aa ü

$= ïïïï Þ " =ýïï= ïïþM

Every finitely forcible graphon is extremal:

minimize 21 1

1

( ( , ) )a=

-åm

j

t F U

Every unique extremal graphon is finitely forcible.??? Every extremal graph problem has a finitely forcible extremal graphon ???

May 2012 34

Finitely forcible graphons

2

3

2( , )32( , )9

t K W

t K W

ìïïïïïíï =î

=

ïïïïGoodman

1/22

4

1( , )21( , )

16

t K W

t C W

=

=

ìïïïïïíïïïïïî

Graham-Chung-Wilson

May 2012 35

Many finitely forcible graphons

Stepfunctions finite graphs with node and edgeweights

Stepfunction:

May 2012 36

L – V.T.Sós

Many finitely forcible graphons

2,1 2

( , ) 0

1( , ) ( , )6

t W

t K W t K W

ìïïïïïïíïï-

î

=

=ïïïï

( , ) 0p x y > p monotone decreasingsymmetric polynomial

finitely forcible

?

January 2011 37


( , ) 0t W = Þ Sp(x,y)=0

( )1 1 1, 1( , ) ( , )a b a ba b

S S

t K W ab x y dx dy b x y n x y e dx- - -

¶= = ×ò ò

( )21 2 ,( , ) ( , ) ( ) ( , )

ii iS

a bx y p x y n x y e e ds t K Wa¶

× + =åò

Stokes( )1 2

1, , 1

( , ) ( )

( 1) ( , ) ( 1) ( , )

a b

a b a b

S

x y n x y e e ds

a t K W b t K W+ +

¶× +

= + + +ò

January 2011 38


May 2012 39

Not too many finitely forcible graphons

Finitely forcible graphons form a set of first category

in (W0,).

May 2012 40

Finitely forcible graphons: conjectures

??? Finitely forcible space of “rows” has finite dimension ???

??? Finitely forcible algebra of k-labeled quantum graphs mod W is finitely generated ???

W=1 iff angle <π/2

??? Is this graphon finitely forcible? ???

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Dense graph limit theory: Extremal graph theory László Lovász Eötvös Loránd University, Budapest…