Local and global convergence

in bounded degree graphs

László Lovász

Eötvös Loránd University, Budapest

Joint work with Christian Borgs,Jennifer Chayes and Jeff Kahn

December 2009 1

Dedicated to the Memory of Oded Schramm

December 2009 2

The Benjamini-Schramm limit

G: simple graph with all degrees ≤ D

BG(v,r)= {nodes at distance ≤ r from node v}

v random uniform node BG(v,r) random graph in Ar

PG(A)= P(BG(v,r)≈A)

Ar= {simple rooted graphs with all degrees ≤ D and radius ≤r }

(G1,G2,…) convergent: is convergent for all A( )nGP A

li( ) m ( )

nGnPP AA

December 2009 3

The Benjamini-Schramm limit

A1

A2

A3

' childof

( ) ( ')A A

n nG GP A P A ' childof

( ) ( ')A A

P A P A

…

December 2009 4

The Benjamini-Schramm limit

= {maximal paths from } = {rooted countable graphs with degrees ≤D}

A = {maximal paths through A}

A = {-algebra generated by the A}

P: probability measure on (,A)

P has some special properties…

December 2009 5

Other limit constructions

December 2009 6

Other limit constructions

?

December 2009 7

Other limit constructions

Measure preserving graph: G=([0,1],E)

(a) all degrees ≤D

(b) X[0,1] Borel N(X) is Borel

( ) ( )X Y

N z Y dz N z X dz(c) X,Y[0,1] Borel

R.Kleinberg – L

December 2009 8

Other limit constructions

Graphing: G=([0,1],E)

Elek

1 1 0 1

0 1 1

[ , ],

measure preserving involuti

,..., , ,...,

:

E= (x, ( ))

o

: , , ,...,

nk k

i i i

i

A A B B

A B

x x i k

December 2009 9

Homomorphism functions

: # of homomorphisms ohom( n o, f t) iGG H H

Weighted version:

( , , , ) : :, ,¡ ¡H V EV E

( ): ( ) ( )

(()

)( )( )

:hom( , ) i jij E

ii V G GV G V H

G H

| ( )|

hom( , )

|(

), )

( |V G

G H

V Ht G H Probability that random map

V(G)V(H) is a hom

December 2009 10

Homomorphism functions

hom( , ) # of -colorings of=qG K q G

3 6hom( , ) # of triangles ofK G G

Examples:

hom(G, ) = # of independent sets in G

December 2009 11

Homomorphism functions

We know ( ) " Î A rGP A A

we know hom( , )

with" £F G

F F rG

we know inj( , )

with" £F G

F F rG

we know ind( , )

with" £F G

F F rG

we know ind( , , )

with and prescribed

degrees ( )

" £

£

F d G rF F

G Dd i D

we know

1/

( ) ,æ ö÷ç" Î £ ÷ç ÷çè ø

A s

D

Gr

P A A sD

December 2009 12

Homomorphism functions

1 2

hom( , ), ,... convergent convergent n

n

F GG G F

G

December 2009 13

Left and right convergence

F HG® ®

very large graphcounting edges,triangles,...spectra,...

counting colorations,stable sets,...statistical physics,...maximum cut,...

December 2009 14

Left and right convergence

1 2

hom( , ), ,... convergent convergent n

n

F GG G F

G

1 2

1/, ,... convergent hom( , ) convergent n

nGG G G H H?

ln ( , )

convergent n

n

t G HH

G

December 2009 15

Examples

2

2, if is even,(C , )

0, otherwisen

nt K

December 2009 16

Examples

ln hom( , )

converges (n,m )n mP P H

nm

4 4P P

hom( , ) hom( , )hom( , )k n m k m n mP P H P P H P P H

Fekete’s Lemma convergence

December 2009 17

Examples

ln hom( , )

converges ( , , even)n mC P Hn m n

nm

7 4C P

December 2009 18

Examples

2 2hom( , ) hom( , )n m n mC P H P P H

2

12 2

hom( , )hom( , )

hom( , )n m

n mm

P P HC P H

P H

2 2 1ln hom( , ) ln hom( , ) ln hom(

ln hom(

, )

2 2, )

n m n m n

m

mP P H C P H P P H

nm nP

nmH

m

nm

December 2009 19

Examples

ln hom( , )

converges ( , ,

connected nonbipartite)

n mC P Hn m

nmH

7 4C P

December 2009 20

Examples

hom( , ) tr( )nn m GC P H A

Construct auxiliary graph G:

( ) homomorphisms

( ) : ( ) ( ) ( )

mV G P H

E G x x E H x

hom( , ) nn m GP P H A J

H connected nonbipartite G connected nonbipartite

ln( ) ln(tr( ))n nG GA J A

December 2009 21

Examples

ln hom( , )( , ) converges if either

, are even, or

is connected nonbipartite

n mC C Hn m

nmn m

H

December 2009 22

Left and right convergence

1 2

ln ( , )convergent

1wit

, ,... co

h min deg( ) 12

nver nt

0

ge n

n

Gt G H

G

H

G

H HD

1 2

ln ( , )convergent

with min deg(

, ,... converge

)

nt

1

n

n

D

t G H

G

H H

G

H

G

December 2009 23

Analogy: the dense case

Left-convergence (homomorphisms from “small” graphs)

Right-convergence (homomorphisms into “small” graphs)

Distance of two graphs (optimal overlay; convergentCauchy)

Limit objects (2-variable functions)

Approximation by bounded-size graphs (Szemerédi Lemma, sampling)

Parameters “continuous at infinity” (parameter testing)

December 2009

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F WÎ

= Õò

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

Limit objects

24

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

Borgs, Chayes,L,Sós,Vesztergombi

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

nG W®

December 2009 25

Limit objects

LS

Conversely, for every graphon W there is

a graph sequence (Gn) such that nG W® LS

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

December 2009 26

Amenable (hyperfinite) limits

o(n) edges

(n) nodes

Small cut decomposition:

December 2009 27

Amenable (hyperfinite) limits

{G1,G2,…} amenable (hyperfinite):

0 ( ), ,

.n n n

n

k G E G X G

connected component of G X has k nodes

X

Can be decomposed into bounded piecesby small cut decomposition.

December 2009 28

Amenable graphs and hyperfinite limits

For a convergent graph sequence,hyperfiniteness is reflected by the limit.

Schramm

Every minor-closed property is testable forgraphs with bounded degree.

Benjamini-Schramm-Shapira

December 2009 29

Regularity Lemma?

-homogeneous: small cut decomposition, each pieceH satisfies

Every sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous piecesby small cuts.

: ( ) ( )H GA P A P A

Elek – LippnerAngel - Szegedy

December 2009 30

Regularity Lemma?

Easy observation:

For every r,D1 and 0 there is a q(r,,D)

such that for every graph G with degrees D

there is a graph H with degrees D and with q nodes

such that for all for all connected graphs F with r nodes

hom( , ) hom( , )F G F H

G H

Alon

A construction for H? Effective bound on q?