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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?