Random graphs

and limits of graph sequences

László Lovász

Microsoft Research

lovasz@microsoft.com

W-random graphs

{ }2: [0,1] symmetric, bounded, measurableW= ®W ¡

{ }0 : : 0 1f f= Î £ £W W

0 1, ,..., [0,1]Fix let iid uniformnW X XÎ ÎW

{1,..., }

( ,

( ( , ))

( , ) )( )( )P i j

V n W

i

n

W X Xj E n W

=

Î =

G

G

Adjacency matrix of weighted graph G, viewed as a function in 0:

GG Wa

WG-random graphs

generalized random graphs

with model G

( ) ( )[0,1]

(( , ,) : )V F

i jij E F

W x x dxt F WÎ

= Õò

( , )

( (

)

)

( ,

)P(random map preserves edges)Gt Ft F G W

V F V G

=

®

=

( , ( , )) ( , ) a.s.t F W n t F W®G

density of F in W

Convergent graph sequences

( , ) ( )simple graph nF t F G t F" ®(Gn) is convergent:

Examples: Paley graphs (quasirandom) half-graphs

closest neighbor graphs ...

Does a convergent graph sequence have a limit?

For every convergent (Gn)

there is a function W0 such that( , ) ( , )nt F G t F W®

B.Szegedy-L

GnW

half-graphs ®

1 12 2( , )n ®G a.s.

( , )n W W®G a.s.

Uniqueness of the limit Borgs-Chayes-L

(( , ) ( ( ): ), )Wx xW yyj j j=

1 2

0

1 2

: ( , ) ( , )

, :[0,1] [0,1]

,

measure preserving

F t F W t F W

W

W W W Wj y

j y

" = Þ

$ Î $ ®

= =

W

W W W

W W W

W W W

W W

WWW

A random graph

with 100 nodes and 2500 edges 1/2

Quasirandom converges to 1/2

Growing uniform attachment graph

If there are n nodes

- with prob c/n, a new node is added,

- with prob (n-c)/n, a new edge is added.

| ( ) |1| ( ) |

2n

n

V GE G

c

æ ö÷ç ÷» ç ÷ç ÷çè ø

A growing

uniform attachment graph

with 200 nodes and 10000 edges

1 max( , )x y-

Fixed preferential attachment graph

Fix n nodes

For m steps

choose 2 random nodes independently

with prob proportional to (deg+1)

and connect them

A preferential attachment graph

with 100 fixed nodes

and with 5,000 (multiple) edges

A preferential attachment graph

with 100 fixed nodes ordered by degrees

and with 5,000 edges

ln( ) ln( )x y

Moments1-variable functions 2-variable functions

[0,1]

( , ) : ( )kt k f f x dx= ò( ) ( )[0,1]

( , ) : ( , )V F

i jij E F

t F W W x x dxÎ

= Õò

These are independentquantities.

These are independentquantities.

Erdős-L-Spencer

Moments determine thefunction up to measure preserving transformation.

Moment sequences are characterized by semidefiniteness

Moments determine thefunction up to measure preserving transformation.

Borgs-Chayes-L

Moment graph parameters are characterized by semidefiniteness

L-Szegedy

Except for multiplicativity over disjoint union:

1 2 1 2( , ) ( , ) ( , )t F F W t F W t F WÈ =

k-labeled graph: k nodes labeled 1,...,k

Connection matrix of graph parameter f

1 21 2() )( , F F fk Ff FM =

1 2

1 2

1 2 ,

, :

:

-labeled graphs

labeled nodes identified

k

F F

F F

F F

Connection matrices

k=2:

...

...

( )f

f is a moment parameter

1( ) 1,

( ,

( ) lim ( ,

)

)n

f K f

M

G

f

f F t F

k

Û

=

Û

= multiplicative

positive semidefinite

L-Szegedy

Gives inequalities between subgraph densities

extremal graph theory

f is reflection positive

Kruskal-Katona Theorem for triangles: 3/ 2( ) ( )t t

Turán’s Theorem for triangles: ( ) ( )(2 ( ) 1)t t t

4

| ( )|( )( )

( )E Ft p

F t F pt p

Graham-Chung-Wilson Theorem about quasirandom graphs:

Extremal graph theory as properties of Ît T

k=2

2 3( ) ( ) ( ) ( ) ( ) ( )t t t t t t

Proof of Kruskal-Katona

Moments1-variable functions 2-variable functions

[0,1]

( , ) : ( )kt k f f x dx= ò( ) ( )[0,1]

( , ) : ( , )V F

i jij E F

t F W W x x dxÎ

= Õò

These are independentquantities.

These are independentquantities.

Erdős-L-Spencer

Moments determine thefunction up to measure preserving transformation.

Moment sequences are characterized by semidefiniteness

Moments determine thefunction up to measure preserving transformation.

Borgs-Chayes-L

Moment graph parameters are characterized by semidefiniteness

L-Szegedy

Moment sequences areinteresting

Moment graph parameters are interesting

( , ) ( , )

( ) ( )Gt F G t F W

V F V G

= =

®P(random map preserves edges)

| ( )| ( , )n

V FKn t F W n F= #(proper -colorings of )

partition functions, homomorphism functions,...

| ( )|2 ,cos(2 ( ))( )E G t F x y Fp - = # eulerian orientations of

L-Szegedy

The following are cryptomorphic:

functions in 0 modulo measure preserving transformations

reflection positive and multiplicative graph parameters f with f(K1)=1

random graph models (n) that are- label-independent- hereditary- independent on disjoint subsets

countable random graphs that are- label-independent- independent on disjoint subsets

Rectangle norm:

,sup (: , )S T

S T

W x y dx dyW´

= òX

Rectangle distance:

1 2, :[0,1] [0,1]

1 2( , ) : infmeasure preserving

WW W Wj y

j yd

®-=X

( )0 0 ,: /d d== WW XX X

The structure of 0

1 21 2( , ) ( ,: )G GW WG G dd = XX 1 2

1 2

( , ) 0

( , ) ( , )

W W

F t F W t F W

d = Û

" =X

Weak Regularity Lemma:

21/0 2

( , ) .

W U

W U

ee

d e

" Î " > $ £

£

stepfunction with steps

such that

WX

X

22/0 2

( , ) .G

W G

W W

ee

d e

" Î " > $ £

£

graph with nodes

such that

WX

X

is compactWXL-Szegedy

Frieze-Kannan

For a sequence of graphs (Gn), the following are equivalent:

(i)

(iii)

(iii)

( , )nt F G F"is convergent

( )nGW is convergent in WX

( ) is Cauchy with respect to nG dX

uniform attachment graphs 1 max( , )x y® -

preferential attachment graphs ln( ) ln( )x y®

random graphs 1/ 2®

Approximate uniqueness

1 2 1 2( , ) ( , ) ( ) ( , )t F W t F W E F W Wd- £ X

Borgs-Chayes-L-T.Sós-Vesztergombi

1 2

1 2

2 24/ 8/| ( ) | 2 ( , ) ( , ) 2

( , )

F V F t F W t F W

W W

e e

d e

-" £ - £

Þ £

with

X

If G1 and G2 are graphs on n nodes so that for all F with

then G1 and G2 can be overlayed so that for all

1 2

2 24/ 8/| ( ) | 2 ( , ) ( , ) 2V F t F G t F Ge e-£ - £

1 2

2( , ) ( , )G Ge S T e S T ne- £

1, ( )S T V GÍ

Local testing for global properties

What to ask?

-Does it have an even number of nodes?

-Is it connected?

-How dense is it (average degree)?

For a graph parameter f, the following are equivalent:

(i) f can be computed by local tests

(ii) ( ) ( )n nG f GÞconvergent convergent

(iii) f is unifomly continuous w.r.t dX

Density of maximum cut is testable.

Borgs-Chayes-L-T.Sós-Vesztergombi

Key fact: 10

( , ),log

( )n W Wn

d <X G

10( , ),

log( )Gn W G

nd <X G