The mathematical challenge

of large networks

László Lovász

Eötvös Loránd University, Budapest

Joint work with Christian Borgs, Jennifer Chayes,Balázs Szegedy, Vera Sós and Katalin Vesztergombi

May 2012 1

Minimize x3-6x over x0

minimum is not attainedin rationals

Minimize 4-cycle density in graphs with edge-density 1/2

minimum is not attainedamong graphs

always >1/16,arbitrarily close for random

graphs

Real numbers are useful

Graph limits are useful

May 2012 2

Graph limits: Why are they needed?

Two extremes:

- dense (cn2 edges)

- bounded degree

well developedgood warm-up case

How dense is the graph?

May 2012 3

Inbetween???Bollobas-RiordanChungConlon-Fox-Zhao

less developed, more difficultmost applications

4May 2012

How is the graph given?

- Graph is HUGE.

- Not known explicitly, not even the number of nodes.

Idealize: define minimum amount of info.

May 2012 5

Dense case: cn2 edges.

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

between sampled nodes.

Bounded degree (d)

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

How is the graph given?

„Property testing”: Arora-Karger-Karpinski,

Goldreich-Goldwasser-Ron, Rubinfeld-Sudan,

Alon-Fischer-Krivelevich-Szegedy, Fischer,

Frieze-Kannan, Alon-Shapira

May 2012 6

Lecture plan

Want to construct completion of the set of graphs.

- What is the distance of two graphs?

- Which graph sequences are convergent?

- How to represent the limit object?

- How does this completion space look like?

- How to approximate graphs? (Regularity Lemmas and sampling)

May 2012 7

Lecture plan

Applications in (dense) extremal graph theory

- Are extremal graph problems decidable?

- Which graphs are extremal?

- Local vs. global extrema

- Is there always an extremal graph?

May 2012 8

Lecture plan

Applications in property testing

- Deterministic and non-deterministic sampling (P=NP)

- Which properties are testable by sampling?

May 2012 9

Lecture plan

The bounded degree case

- Different limit objects: involution-invariant distributions, graphings

- Local algorithms and distributed computing

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

May 2012 10

A random graph with 100

nodes and with 2500 edgesMay 2012 11

Pixel pictures

Rearranging rows and columns

May 2012 12

Pixel pictures

May 2012 13

Pixel pictures

A randomly grown uniform

attachment graph on 200 nodes

At step n: - a new node is born; - any two nodes are

joined with probability 1/n

Ignore multiplicity of edges

Approximation by small: Regularity Lemma

Szemerédi1975

May 2012 14

Nodes can be so ordered

essentially random

Approximation by small: Regularity Lemma

May 2012 15

The nodes of any graph can be partitioned

into a small number

of essentially equal parts

so that

the bipartite graphs between 2 parts

are essentially random

(with different densities).with k2 exceptions

for subsets X,Y of parts Vi,Vj# of edges between X and Y

is pij|X||Y| ± (n/k)2

Given >0

22

21 12 }ke e

£ £N

difference at most 1

Approximation by small: Regularity Lemma

May 2012 16

May 2012 17

Original Regularity Lemma Szemerédi 1976

“Weak” Regularity Lemma Frieze-Kannan 1999

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

Approximation by small: Regularity Lemma

1 max( , )- x y

May 2012 18

A randomly grown uniform

attachment graph on 200 nodes

Graph limits: Examples

Knowing the limit W

knowing many properties (approximately).

Graph limits: Why are they useful?

3[0,1]

( , ) ( , ) ( , )W x y W y z W z x dx dy dzòtriangle density

May 2012 19

May 2012 20

Limit objects: the math

distribution of k-samples

is convergent for all k

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

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

May 2012 21

Limit objects: the math

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"

Randomly grown prefix attachment graph

At step n:- a new node is born;- connects to a random previous node and all its predecessors

May 2012 22

Limit objects: an example

Limit objects: an example

A randomly grown prefix attachment graph

with 200 nodes

Is this graph sequenceconvergent at all?

Yes, by computing subgraph densities!

This tends to some shades of gray; is that the limit?

No, by computing triangle densities!

May 2012 23

A randomly grown prefix attachment graph

with 200 nodes (ordered by degrees)

This also tends to some shades of gray; is that the limit?

No…

May 2012 24

Limit objects: an example

The limit of randomly grown prefix attachment graphs

May 2012 25

Limit objects: an example

26

The distance of two graphs

May 2012

'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

| ( ) | ' | ( ') |V G n n V G= ¹ =(c) blow up nodes

*( , ') lim ( ( '), '( ))k

G G G kn G knd d®¥

=X X

finite definitionby fractional overlay

27May 2012

Examples: 1, 2

1( , (2 , ))

8n nK nXd »G

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

The distance of two graphs

May 2012 28

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

pij: density between Si and Sj

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

“Weak” Regularity Lemma

May 2012 29

1

2( , ) .

log

For every graph and there is a -partition

such that

X

G k k

d G Gk

³

£P

P

Frieze-Kannan

“Weak” Regularity Lemma

May 2012 30

| ( , ) ( , ) | ( ) ( , )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

Counting Lemmas

May 2012 31

Equivalence of convergence notions

A graph sequence (G1,G2,...) is convergent iff

it is Cauchy in .

December 2008 32

'( , ') 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

The distance of two graphons

May 2012 33

The semimetric space (W0,) is compact.

“Strong” Regularity Lemma

May 2012 34

Limit objects: the 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 35

Limit objects: the 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 . Regularity Lemma + martingales L-SzegedyExchangeable random variables Aldous; Diaconis-JansonUltraproducts Elek-Szegedy

May 2012 36

Limit objects: the 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 .

W-random graphs(sampling from a graphon)

May 2012 37

Limit objects: the 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 .

Constructing canonical representationBorgs-Chayes-L

Exchangeable random variablesKallenberg; Diaconis-Janson

Inverse Counting Lemma, measure compactnessBollobas-Riordan

May 2012 38

Thanks, that’sall for today!