regularity partitions and the topology of graphons
DESCRIPTION
Regularity partitions and the topology of graphons. L á szl ó Lov á sz Eötvös Lor ánd University, Budapest. Joint work Bal á zs Szegedy. The Szemerédi Regularity Lemma. given ε >0, # of parts k satisfies 1 / ε k f ( ε ). difference at most 1. with ε k 2 exceptions. - PowerPoint PPT PresentationTRANSCRIPT
Regularity partitions
and the topology of graphons
László Lovász
Eötvös Loránd University, Budapest
Joint work Balázs Szegedy
August 2010 1
August 2010 2
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 pij).with εk2 exceptions
given ε>0, # of parts
k satisfies 1/ ε kf(ε)
difference at most 1
for subsets X,Y of the two parts,# of edges between X and Y
is pij|X||Y| ε(n/k)2
The Szemerédi Regularity Lemma
August 2010 3
Original Regularity Lemma Szemerédi 1976
“Weak” Regularity Lemma Frieze-Kannan 1999
“Strong” Regularity Lemma Alon – Fisher- Krivelevich - M. Szegedy 2000
Tao 2005
L-Szegedy 2006
The Szemerédi Regularity Lemma
Low rank matrix approximation - Frieze-Kannan
August 2010 4
The many facets of the Lemma
Probability, informationtheory - Tao
Approximation theory - L-Szegedy
Compactness - L-Szegedy
Dimensionality - L-Szegedy
Measure theory - Bollobás-Nikiforov
Sparse Regularity Lemma
Gerke, Kohayakawa, Luczak, Rödl, Steger,Hypergraph Regularity
Lemma
Frankl, Gowers, N
agle, Rödl, S
chacht
Arithmetic Regularity Lemma
Green, Tao
Regularity Lemma and ultraproducts
Elek, Szegedy
August 2010
{ }20 : [0,1] [0,1] symmetric, measurableWW = ®
Graphons
5
Could be:
[ ]2
( , , ) :
: 0,1 :
probability space
symmetric, measurable
J A
W J
p
®
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
August 2010 6
August 2010 7
'( , ') ( , )G GG G W WX Xd d=
Cut distance of graphons
, [0,1]sup
S T S T
W W
X cut norm
i( , ') nf 'W WW W
X
cut distance
2
1 0,1W L
measurepreserving
August 2010
Regularity Lemma and cut distance
8
0W WÎP: measurable partition of [0,1],
1( , ) ,
( ) ( )( , ) :
S T
W x W s t ds dt x S yS
y TTP P P
l l ´
Î Î Î= Îò
0
21 : ,
logW k k W W
kPW P: PX
" Î " ³ $ = - £
Weak Regularity Lemma:
is compact0( , )W Xd
Strongest Regularity Lemma:
August 2010
( ) ( )[0,1]
( , )( , )V F
i jij E F
W x x dxt F WÎ
= Õò0W WÎ
Subgraph density
9
( , ) ( , )Gt F W t F G=
hom( , ) : # of homomorphisms of intoFF GG
| ( )|
hom( , )
| ( ) |( , )
V F
F G
V Gt F G Probability that random map
V(F)V(G) is a hom
August 2010
Graphons as limit objects
10
1 2( , ,.. ( ,.) )convergent: is convergentnF t F GG G "
: ( , (: ) , )nn F t F GW t WG F®® "
( , ) 0nG
W WXdÛ ®
Cauchy in the -distanceXdÛ
Borgs, Chayes, L, Sós, Vesztergombi
For every convergent graph sequence (Gn)there is a graphon such that0W Î W
nG W®
August 2010 11
Graphons as limit objects
Conversely, for every graphon W there is
a graph sequence (Gn) such that nG W®L-Szegedy
W is essentially unique (up to measure-preserving transformation).
Borgs-Chayes-L
A randomly grown uniform
attachment graph with 200 nodes1 max( , )x y-
August 2010 12
Example: Randomly growing graphs
August 2010 13
Example: Generalized random graph
Random with density 1/3
Random with density 2/3
Random with density 1/3
August 2010 14
Example: Borsuk graphon
W(x,y)=1(|x-y|>2-d-2)
Gn W
Gn has weak regularity partition with O(d) classes
Neighborhoods in Gn have VC-dimension d+1
W has a d-dimensional „underlying space”
Gn does not contain Fd+1 as an induced subgraph
F3:
Gn: induced subgraph on n random nodes
Sd with uniform distribution
August 2010 15
The topology of a graphon
1: ( ,.) ( ,( , .) ( , )) ( , )=EW vW s W t W sd vs W vt t= - -
: ( , ) ( ,( ) ) ( , ) (, )) ,( EvW x z W z y dz W s v WW W y vx to = =ò
1: ( )( ,.) ( )( ,.)
( ( , ) ( , )) ( ( , )
)
(
,
, )
(
)E E E
W W
v u w
W W s W W t
W s u W u v W
d s t
t w W w v
o o o= -
= -
s
t
v
wu
Squaring the adjacency matrix
August 2010 16
The topology of a graphon
( , , ) : graphonJ WA ,p
( , )WJ d ( , )W WJ d o
Complete metric spacesA = Borel sets has full support
After a lot ofcleaning
CompactNot alwayscompact pure
graphon
August 2010 17
Example: Generalized random graphs again
(J,rG): discrete (J,rW) (J,rWoW):
(J,rGoG): (J,rWoW):
Gromov-Wasserstein convergence
August 2010 18
Example: Borsuk graphon again
( , ) ( , ) ( , )W W Wd x y d x y x yo: : S
Gn: induced subgraph on n random nodes
G’n: randomly delete half of the edges from Gn
''nG
W W=
' ' '( , ) const, ( , ) ( , )W W Wd x y d x y x yo: : S
August 2010 19
(: ( , )) E WW WW xS d d x SSJ o oÍ =
[0,1] : ( ) ( , )partition of d W Wr PP P X=
average ε-net
regular partition
SJ Voronoi cells of S form a partition with
partition P={V1,...,Vk} of [0,1] vi Vi with
( ) 8 ( )W Wr d SP o<
1({ ,..., }) 12 ( )kW Wd v v r Po <
Regularity and dimensionality
August 2010 20
Theorem.
ε>0, the metric space (J,rWoW) can be partitioned into
a set of measure <ε, and sets with diameter <ε.22/2 e
Regularity and dimensionality
August 2010 21
Extremal graph theory and dimensionality
F: bipartite graph with bipartition (U,V), G: graphW: graphon
F bi-induced subgraph of G: U’,V’V(G), disjoint,subgraph formed by edges between U’ and V’
is isomorphic to F
F bi-induced subgraph of W:
[0,1] , ,( ) ( )
( , ) (1 ( , )) 0i j i jU V i U j V i U j V
ij E F ij E F
W x x W x x dxÈ Î Î Î Î
Î Ï
- >Õ Õò
August 2010 22
Extremal graph theory and dimensionality
Theorem.
F is not a bi-induced subgraph of W
W is 0-1 valued, (J,rW) is compact, and has finite packing
dimension.
Key fact: VC-dimension of neighborhoods is bounded
August 2010 23
Extremal graph theory and dimensionality
Corollary.
P: hereditary bigraph property not containing all bigraphs.
(J,W): pure graphon in its closure
W is 0-1 valued, (J,rW ) is compact
and has bounded dimension.
August 2010 24
Extremal graph theory and dimensionality
Corollary.
P: hereditary graph property not containing all graphs,
such that W in its closure is 0-1 valued,
(J,W): pure graphon in its closure
(J,rW ) is compact and has bounded dimension.
August 2010 25
Example
P : triangle-free
PÎ
Corollary.
F is not a bi-induced subgraph of G
>0, G has a weak regularity partition with error
with at most classes.
August 2010 26
Extremal graph theory, dimensionality and regularity
( )10V FFc e-