random graphs and limits of graph sequences lászló lovász microsoft research [email protected]
TRANSCRIPT
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