local and global convergence in bounded degree graphs lászló lovász eötvös loránd university,...

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?