Graph algebras and extremal graph theory László Lovász April 20131.
Random graphs and limits of graph sequences László Lovász Microsoft Research...
Transcript of Random graphs and limits of graph sequences László Lovász Microsoft Research...
W-random graphs
{ }2: [0,1] symmetric, bounded, measurableW= ®W ¡
{ }0 : : 0 1f f= Î £ £W W
0 1, ,..., [0,1]Fix let iid uniformnW X XÎ ÎW
{1,..., }
( ,
( ( , ))
( , ) )( )( )P i j
V n W
i
n
W X Xj E n W
=
Î =
G
G
Adjacency matrix of weighted graph G, viewed as a function in 0:
GG Wa
WG-random graphs
generalized random graphs
with model G
( ) ( )[0,1]
(( , ,) : )V F
i jij E F
W x x dxt F WÎ
= Õò
( , )
( (
)
)
( ,
)P(random map preserves edges)Gt Ft F G W
V F V G
=
®
=
( , ( , )) ( , ) a.s.t F W n t F W®G
density of F in W
Convergent graph sequences
( , ) ( )simple graph nF t F G t F" ®(Gn) is convergent:
Examples: Paley graphs (quasirandom) half-graphs
closest neighbor graphs ...
Does a convergent graph sequence have a limit?
For every convergent (Gn)
there is a function W0 such that( , ) ( , )nt F G t F W®
B.Szegedy-L
GnW
half-graphs ®
1 12 2( , )n ®G a.s.
( , )n W W®G a.s.
Uniqueness of the limit Borgs-Chayes-L
(( , ) ( ( ): ), )Wx xW yyj j j=
1 2
0
1 2
: ( , ) ( , )
, :[0,1] [0,1]
,
measure preserving
F t F W t F W
W
W W W Wj y
j y
" = Þ
$ Î $ ®
= =
W
W W W
W W W
W W W
W W
WWW
Growing uniform attachment graph
If there are n nodes
- with prob c/n, a new node is added,
- with prob (n-c)/n, a new edge is added.
| ( ) |1| ( ) |
2n
n
V GE G
c
æ ö÷ç ÷» ç ÷ç ÷çè ø
Fixed preferential attachment graph
Fix n nodes
For m steps
choose 2 random nodes independently
with prob proportional to (deg+1)
and connect them
A preferential attachment graph
with 100 fixed nodes ordered by degrees
and with 5,000 edges
ln( ) ln( )x y
Moments1-variable functions 2-variable functions
[0,1]
( , ) : ( )kt k f f x dx= ò( ) ( )[0,1]
( , ) : ( , )V F
i jij E F
t F W W x x dxÎ
= Õò
These are independentquantities.
These are independentquantities.
Erdős-L-Spencer
Moments determine thefunction up to measure preserving transformation.
Moment sequences are characterized by semidefiniteness
Moments determine thefunction up to measure preserving transformation.
Borgs-Chayes-L
Moment graph parameters are characterized by semidefiniteness
L-Szegedy
Except for multiplicativity over disjoint union:
1 2 1 2( , ) ( , ) ( , )t F F W t F W t F WÈ =
k-labeled graph: k nodes labeled 1,...,k
Connection matrix of graph parameter f
1 21 2() )( , F F fk Ff FM =
1 2
1 2
1 2 ,
, :
:
-labeled graphs
labeled nodes identified
k
F F
F F
F F
Connection matrices
f is a moment parameter
1( ) 1,
( ,
( ) lim ( ,
)
)n
f K f
M
G
f
f F t F
k
Û
=
Û
= multiplicative
positive semidefinite
L-Szegedy
Gives inequalities between subgraph densities
extremal graph theory
f is reflection positive
Kruskal-Katona Theorem for triangles: 3/ 2( ) ( )t t
Turán’s Theorem for triangles: ( ) ( )(2 ( ) 1)t t t
4
| ( )|( )( )
( )E Ft p
F t F pt p
Graham-Chung-Wilson Theorem about quasirandom graphs:
Extremal graph theory as properties of Ît T
Moments1-variable functions 2-variable functions
[0,1]
( , ) : ( )kt k f f x dx= ò( ) ( )[0,1]
( , ) : ( , )V F
i jij E F
t F W W x x dxÎ
= Õò
These are independentquantities.
These are independentquantities.
Erdős-L-Spencer
Moments determine thefunction up to measure preserving transformation.
Moment sequences are characterized by semidefiniteness
Moments determine thefunction up to measure preserving transformation.
Borgs-Chayes-L
Moment graph parameters are characterized by semidefiniteness
L-Szegedy
Moment sequences areinteresting
Moment graph parameters are interesting
( , ) ( , )
( ) ( )Gt F G t F W
V F V G
= =
®P(random map preserves edges)
| ( )| ( , )n
V FKn t F W n F= #(proper -colorings of )
partition functions, homomorphism functions,...
| ( )|2 ,cos(2 ( ))( )E G t F x y Fp - = # eulerian orientations of
L-Szegedy
The following are cryptomorphic:
functions in 0 modulo measure preserving transformations
reflection positive and multiplicative graph parameters f with f(K1)=1
random graph models (n) that are- label-independent- hereditary- independent on disjoint subsets
countable random graphs that are- label-independent- independent on disjoint subsets
Rectangle norm:
,sup (: , )S T
S T
W x y dx dyW´
= òX
Rectangle distance:
1 2, :[0,1] [0,1]
1 2( , ) : infmeasure preserving
WW W Wj y
j yd
®-=X
( )0 0 ,: /d d== WW XX X
The structure of 0
1 21 2( , ) ( ,: )G GW WG G dd = XX 1 2
1 2
( , ) 0
( , ) ( , )
W W
F t F W t F W
d = Û
" =X
Weak Regularity Lemma:
21/0 2
( , ) .
W U
W U
ee
d e
" Î " > $ £
£
stepfunction with steps
such that
WX
X
22/0 2
( , ) .G
W G
W W
ee
d e
" Î " > $ £
£
graph with nodes
such that
WX
X
is compactWXL-Szegedy
Frieze-Kannan
For a sequence of graphs (Gn), the following are equivalent:
(i)
(iii)
(iii)
( , )nt F G F"is convergent
( )nGW is convergent in WX
( ) is Cauchy with respect to nG dX
uniform attachment graphs 1 max( , )x y® -
preferential attachment graphs ln( ) ln( )x y®
random graphs 1/ 2®
Approximate uniqueness
1 2 1 2( , ) ( , ) ( ) ( , )t F W t F W E F W Wd- £ X
Borgs-Chayes-L-T.Sós-Vesztergombi
1 2
1 2
2 24/ 8/| ( ) | 2 ( , ) ( , ) 2
( , )
F V F t F W t F W
W W
e e
d e
-" £ - £
Þ £
with
X
If G1 and G2 are graphs on n nodes so that for all F with
then G1 and G2 can be overlayed so that for all
1 2
2 24/ 8/| ( ) | 2 ( , ) ( , ) 2V F t F G t F Ge e-£ - £
1 2
2( , ) ( , )G Ge S T e S T ne- £
1, ( )S T V GÍ
Local testing for global properties
What to ask?
-Does it have an even number of nodes?
-Is it connected?
-How dense is it (average degree)?
For a graph parameter f, the following are equivalent:
(i) f can be computed by local tests
(ii) ( ) ( )n nG f GÞconvergent convergent
(iii) f is unifomly continuous w.r.t dX
Density of maximum cut is testable.
Borgs-Chayes-L-T.Sós-Vesztergombi