Regularity partitions

and the topology of graphons

László Lovász

Eötvös Loránd University, Budapest

Joint work Balázs Szegedy

August 2010 1

August 2010 2

The nodes of graph can be partitioned

into a bounded number

of essentially equal parts

so that

almost all bipartite graphs between 2 parts

are essentially random

(with different densities pij).with εk2 exceptions

given ε>0, # of parts

k satisfies 1/ ε kf(ε)

difference at most 1

for subsets X,Y of the two parts,# of edges between X and Y

is pij|X||Y| ε(n/k)2

The Szemerédi Regularity Lemma

August 2010 3

Original Regularity Lemma Szemerédi 1976

“Weak” Regularity Lemma Frieze-Kannan 1999

“Strong” Regularity Lemma Alon – Fisher- Krivelevich - M. Szegedy 2000

Tao 2005

L-Szegedy 2006

The Szemerédi Regularity Lemma

Low rank matrix approximation - Frieze-Kannan

August 2010 4

The many facets of the Lemma

Probability, informationtheory - Tao

Approximation theory - L-Szegedy

Compactness - L-Szegedy

Dimensionality - L-Szegedy

Measure theory - Bollobás-Nikiforov

Sparse Regularity Lemma

Gerke, Kohayakawa, Luczak, Rödl, Steger,Hypergraph Regularity

Lemma

Frankl, Gowers, N

agle, Rödl, S

chacht

Arithmetic Regularity Lemma

Green, Tao

Regularity Lemma and ultraproducts

Elek, Szegedy

August 2010

{ }20 : [0,1] [0,1] symmetric, measurableWW = ®

Graphons

5

Could be:

[ ]2

( , , ) :

: 0,1 :

probability space

symmetric, measurable

J A

W J

p

®

G

0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0

AG

WG

Pixel pictures

August 2010 6

August 2010 7

'( , ') ( , )G GG G W WX Xd d=

Cut distance of graphons

, [0,1]sup

S T S T

W W

X cut norm

i( , ') nf 'W WW W

X

cut distance

2

1 0,1W L

measurepreserving

August 2010

Regularity Lemma and cut distance

8

0W WÎP: measurable partition of [0,1],

1( , ) ,

( ) ( )( , ) :

S T

W x W s t ds dt x S yS

y TTP P P

l l ´

Î Î Î= Îò

0

21 : ,

logW k k W W

kPW P: PX

" Î " ³ $ = - £

Weak Regularity Lemma:

is compact0( , )W Xd

Strongest Regularity Lemma:

August 2010

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F WÎ

= Õò0W WÎ

Subgraph density

9

( , ) ( , )Gt F W t F G=

hom( , ) : # of homomorphisms of intoFF GG

| ( )|

hom( , )

| ( ) |( , )

V F

F G

V Gt F G Probability that random map

V(F)V(G) is a hom

August 2010

Graphons as limit objects

10

1 2( , ,.. ( ,.) )convergent: is convergentnF t F GG G "

: ( , (: ) , )nn F t F GW t WG F®® "

( , ) 0nG

W WXdÛ ®

Cauchy in the -distanceXdÛ

Borgs, Chayes, L, Sós, Vesztergombi

For every convergent graph sequence (Gn)there is a graphon such that0W Î W

nG W®

August 2010 11

Graphons as limit objects

Conversely, for every graphon W there is

a graph sequence (Gn) such that nG W®L-Szegedy

W is essentially unique (up to measure-preserving transformation).

Borgs-Chayes-L

A randomly grown uniform

attachment graph with 200 nodes1 max( , )x y-

August 2010 12

Example: Randomly growing graphs

August 2010 13

Example: Generalized random graph

Random with density 1/3

Random with density 2/3

Random with density 1/3

August 2010 14

Example: Borsuk graphon

W(x,y)=1(|x-y|>2-d-2)

Gn W

Gn has weak regularity partition with O(d) classes

Neighborhoods in Gn have VC-dimension d+1

W has a d-dimensional „underlying space”

Gn does not contain Fd+1 as an induced subgraph

F3:

Gn: induced subgraph on n random nodes

Sd with uniform distribution

August 2010 15

The topology of a graphon

1: ( ,.) ( ,( , .) ( , )) ( , )=EW vW s W t W sd vs W vt t= - -

: ( , ) ( ,( ) ) ( , ) (, )) ,( EvW x z W z y dz W s v WW W y vx to = =ò

1: ( )( ,.) ( )( ,.)

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

)

(

,

, )

(

)E E E

W W

v u w

W W s W W t

W s u W u v W

d s t

t w W w v

o o o= -

= -

s

t

v

wu

Squaring the adjacency matrix

August 2010 16

The topology of a graphon

( , , ) : graphonJ WA ,p

( , )WJ d ( , )W WJ d o

Complete metric spacesA = Borel sets has full support

After a lot ofcleaning

CompactNot alwayscompact pure

graphon

August 2010 17

Example: Generalized random graphs again

(J,rG): discrete (J,rW) (J,rWoW):

(J,rGoG): (J,rWoW):

Gromov-Wasserstein convergence

August 2010 18

Example: Borsuk graphon again

( , ) ( , ) ( , )W W Wd x y d x y x yo: : S

Gn: induced subgraph on n random nodes

G’n: randomly delete half of the edges from Gn

''nG

W W=

' ' '( , ) const, ( , ) ( , )W W Wd x y d x y x yo: : S

August 2010 19

(: ( , )) E WW WW xS d d x SSJ o oÍ =

[0,1] : ( ) ( , )partition of d W Wr PP P X=

average ε-net

regular partition

SJ Voronoi cells of S form a partition with

partition P={V1,...,Vk} of [0,1] vi Vi with

( ) 8 ( )W Wr d SP o<

1({ ,..., }) 12 ( )kW Wd v v r Po <

Regularity and dimensionality

August 2010 20

Theorem.

ε>0, the metric space (J,rWoW) can be partitioned into

a set of measure <ε, and sets with diameter <ε.22/2 e

Regularity and dimensionality

August 2010 21

Extremal graph theory and dimensionality

F: bipartite graph with bipartition (U,V), G: graphW: graphon

F bi-induced subgraph of G: U’,V’V(G), disjoint,subgraph formed by edges between U’ and V’

is isomorphic to F

F bi-induced subgraph of W:

[0,1] , ,( ) ( )

( , ) (1 ( , )) 0i j i jU V i U j V i U j V

ij E F ij E F

W x x W x x dxÈ Î Î Î Î

Î Ï

- >Õ Õò

August 2010 22

Extremal graph theory and dimensionality

Theorem.

F is not a bi-induced subgraph of W

W is 0-1 valued, (J,rW) is compact, and has finite packing

dimension.

Key fact: VC-dimension of neighborhoods is bounded

August 2010 23

Extremal graph theory and dimensionality

Corollary.

P: hereditary bigraph property not containing all bigraphs.

(J,W): pure graphon in its closure

W is 0-1 valued, (J,rW ) is compact

and has bounded dimension.

August 2010 24

Extremal graph theory and dimensionality

Corollary.

P: hereditary graph property not containing all graphs,

such that W in its closure is 0-1 valued,

(J,W): pure graphon in its closure

(J,rW ) is compact and has bounded dimension.

August 2010 25

Example

P : triangle-free

PÎ

Corollary.

F is not a bi-induced subgraph of G

>0, G has a weak regularity partition with error

with at most classes.

August 2010 26

Extremal graph theory, dimensionality and regularity

( )10V FFc e-