July 2009 1

The Mathematical Challenge

of Large Networks

László Lovász

Eötvös Loránd University, Budapest

lovasz@cs.elte.hu

July 2009 2

What properties to study?

- Does it have an even number of nodes?

- How dense is it (average degree)?

- Is it connected?

- What are the connected components?

Very large graphs Questions

July 2009 3

Very large graphs Framework

How to obtain information about them?

- Graph is HUGE.

- Not known explicitly, not even number of nodes.

July 2009 4

Very large graphs Dense framework

How to obtain information about them?

Dense case: cn2 edges.

- We can sample a uniform random node a bounded number of times, and see edges

between sampled nodes.

July 2009 5

Very large graphs Sparse framework

How to obtain information about them?

Sparse case: Bounded degree (d).

- We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth.

July 2009 6

Very large graphs Alternatives

How to obtain information about them?

- Observing global process locally, for some time.

- Observing global parameters (statistical physics).

July 2009 7

Very large graphs Models

How to model them?

Erdős-Rényi random graphs

Albert-Barabási graphs

Many other randomly growing models

July 2009 8

Very large graphs Approximations

How to approximate them?

- By smaller graphs

Szemerédi partitions (regularity lemmas)

- By larger graphsGraph limits

July 2009 9

Very large dense graphs Distance

How to measure their distance?

'2, ( )

| ( , ) ( , ) |( , ') max G G

S T V G

e S T e S Td G G

nÍ

-=X

( ) ( ')V G V G=(a)

(b) | ( ) | | ( ') |V G V G n= =

*

'( , ') min ( , ')

G GG G d G Gd

«=X X

cut distance

July 2009 10

Very large dense graphs Distance

How to measure their distance?

| ( ) | ' | ( ') |V G n n V G= ¹ =(c)

blow up nodes, or fractional overlay

(( ), ( ')

) ( ')

10

'(

1,)ij i V G j V G ij ij

i V G j V G

X Xn n

X ÎÎ Î

Î =³ =å å

( ) ( , ( ) , ( ')')min max

( , ')

( ' )X S V G V G uviu jv ij

i j V G u v V G

G G

X X a a

d

Í ´ Î Î

=

= -å å

X

July 2009 11

Very large dense graphs Distance

Examples: 1, 2

1( , (2 , ))

8n nK nXd »G

1 11 22 2( , ), ( , ) 1)( ) (n n od =X G G

11 2( , ), 1)( ) (X n od =G

1/2

July 2009 12

Very large dense graphs Sampling Lemmas

( ) ( '), ( ), ,, ' : graphs with randomV G V G VG kG GS S= Í =

10( , [ ])

logG G S

kd <X

With large probability,

Borgs-Chayes-Lovász-Sós-Vesztergombi

1/ 4

10( [ ], '[ ]) ( , ')d G S G S d G G

k- <X X

With large probability,

Alon-Fernandez de la Vega-Kannan-Karpinski+

July 2009 13

Approximating by smaller Regularity Lemmas

Original Regularity Lemma Szemerédi 1976

“Weak” Regularity Lemma Frieze-Kannan 1999

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

July 2009 14

Approximating by smaller Regularity Lemmas

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).with εk2 exceptions

given ε>0, # of parts kis between 1/ ε and f(ε)

difference at most 1

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

is p|X||Y| εn2

July 2009 15

Approximating by smaller Regularity Lemmas

“Weak” Regularity Lemma (Frieze-Kannan):

{ }1,...,partition = kS SP

pij: density between Si and Sj

GP: complete graph on V(G) with edge weights pij

July 2009 16

Approximating by smaller Regularity Lemmas

1

2( , ) .

log

For every graph and there is a -partition

such that

X

G k k

d G Gk

³

£P

P

“Weak" Regularity Lemma (Frieze-Kannan):

July 2009 17

“Weak” Regularity Lemma Similarity distance

2 :( , ) ( ) ( )E E Ev u su vu w tw wvd s t a a a a= -

Fact 1. This is a metric, computable by sampling

Fact 2. Weak Szemerédi partition partition most nodes

into sets with small diameter

LL – B. Szegedy

July 2009 18

“Weak” Regularity Lemma Algorithm

- Select random nodes v1, v2, ...

- Put vi in U iff d2(vi,u)>ε for all uU.

- Begin with U=.

- Stop if for more than 1/ε2 trials, U did not grow.

Algorithm to construct representatives of classes:

size bounded by f(ε)

July 2009 19

“Weak” Regularity Lemma Algorithm

Let U={u1,...,uk}.

Put a node v in Vi iff i is the first index with d2(ui,v)ε.

Algorithm to decide in which class does v belong:

July 2009 20

Max Cut in huge graphs Sampling?

cut with many edges

July 2009 21

Approximating by larger Convergent graph sequences

(i) and (ii) are equivalent.

1 2( , ,...) ( , )convergent: is convergentnG G F t F G"(ii)

(G1, G2,...) convergent: Cauchy in the -metric.dX(i)

distribution of k-samplesis convergent for all k

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

July 2009 22

Approximating by larger Convergent graph sequences

| ( , ) ( , ) | ( ) ( , )t F G t F H E F G Hd- £ X"Counting lemma":

1| ( , ) ( , ) |t F G t F H

k- £“Inverse counting lemma": if

10( , )

logG H

kd <X

for all graphs F with k nodes, then

(i) and (ii) are equivalent.

A random graph

with 100 nodes and with 2500 edges1/2

July 2009 23

Rearranging the rows and columns

July 2009 24

Approximation by small: Szemerédi's Regularity Lemma

July 2009 25

Approximation by small: Szemerédi's Regularity Lemma

Nodes can be so labeled essentially randomJuly 2009 26

A randomly grown

uniform attachment graph

with 200 nodes

1 max( , )x y-

Approximation by infinite

July 2009 27

December 2008 28

Approximating by larger Limit objects

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

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F WÎ

= Õò

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

(graphon)

December 2008 29

Approximating by larger Limit objects

Distance of functions

'( , ') inf ( , ')

XX W W

d W WW W

'( , ') ( , )X X G GG G W Wd d=

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

XS T S T

W Wd W W

December 2008 30

Approximating by larger Limit objects

( , ) 0(i)nGW Wd ®

( ) ( , ) ( , )(ii) nF t F G t F W" ®

Converging to a function

:nG W®

(i) and (ii) are equivalent.

December 2008 31

Approximating by larger Limit objects

LL – B. Szegedy

For every convergent graph sequence (Gn)

there is a such that .

Conversely, W (Gn) such that

0W Î W nG W®

nG W®

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

Borgs – Chayes - LL

Extension to sparse graphs?

July 2009 32

Convergence of graph sequences

Limit objects

Left and right convergence

Dense Bounded degree

Erdős-L-Spencer 1979Borgs-Chayes-L-Sós -Vesztergombi 2006

Aldous 1989Benjamini-Schramm 2001

(graphons) L-B.Szegedy 2006

(graphings) Elek 2007

Benjamini-Schramm 2001

Borgs-Chayes-L-Sós -Vesztergombi 2010?

Borgs-Chayes-Kahn-L 2010?

Extension to sparse graphs?

July 2009 33

Distance

Property testing

Dense Bounded degree

Goldreich-Goldwasser -Ron 1998

Goldreich-Ron 1997Arora-Karger-Karpinski 1995

Borgs-Chayes-L-Sós -Vesztergombi 2008

?

Regularity Lemma

Szemerédi, Frieze-Kannan,Alon-Fischer-Krivelevich-Szegedy, Tao, L-Szegedy,…

?