Random graphs
and limits of graph sequences
László Lovász
Microsoft Research
[email protected]
W-random graphs
W = {W : [0,1]2 ® ¡
W0 :=
{f
symmetric, bounded, measurable}
Î W : 0 £ f £ 1}
Fix W Î W0 , let X 1 ,..., X n Î [0,1] iid uniform
V (G(n,W )) = {1,..., n}
P(ij Î E (G(n,W )) ) = W ( X i , X j )
Adjacency matrix of weighted graph G,
viewed as a function in W0:
WG-random graphs

generalized random graphs
with model G
G a WG
t ( F ,W ) :=
ò
[0,1]V ( F )
Õ
W ( xi , x j ) dx
density of F in W
ij Î E ( F )
t ( F , G ) = t ( F ,WG ) =
P(random map V ( F ) ® V (G ) preserves edges)
t ( F , G(W , n)) ® t ( F , W ) a.s.
Convergent graph sequences
(Gn) is convergent: " simple graph F t ( F , Gn ) ® t ( F )
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 WW0 such that
t ( F , Gn ) ® t ( F ,W )
B.Szegedy-L
G(n, 12 ) ®
1
2
G(n,W ) ® W
half-graphs ®
a.s.
a.s.
GnW
Uniqueness of the limit
W
W
W
W
Borgs-Chayes-L
W
W W W
W W W
W W W
W j ( x, y) := W (j ( x), j ( y))
" F : t ( F ,W1 ) = t ( F ,W2 )
Þ
$W Î W0 $j , y :[0,1] ® [0,1] measure preserving
W1 = W j , W2 = W y
A random graph
with 100 nodes and 2500 edges
Quasirandom  converges to 1/2
1/2
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.
ö
|
V
(
G
)
|
1æ
n
÷
÷
| E (Gn ) |» ç
ç
÷
÷
cç
è 2
ø
A growing
uniform attachment graph
with 200 nodes and 10000 edges
1- max( x, y )
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
and with 5,000 (multiple) edges
A preferential attachment graph
ln( x) ln( y )
with 100 fixed nodes ordered by degrees
and with 5,000 edges
Moments
1-variable functions
t (k , f ) :=
ò
f k ( x) dx
[0,1]
These are independent
quantities.
2-variable functions
t ( F , W ) :=
ò
[0,1]V ( F )
Õ
W ( xi , x j ) dx
ij Î E ( F )
These are independent
quantities.
ErdősLSpencer
Moments determine the
Moments determine the
Borgsfunction up to measure
function up to measure
ChayesExcept
for
multiplicativity
over
disjoint
union:
preserving transformation. preserving transformation. L
t(F
)t ( F2 ,Wgraph
)
1 È F2 , W ) = t ( F1 , W
Moment
sequences
Moment
parameters
Lare characterized by
are characterized by
Szegedy
semidefiniteness
semidefiniteness
Connection matrices
k-labeled graph: k nodes labeled 1,...,k
F1 , F2 : k -labeled graphs
F1 F2 : F1  F2 , labeled nodes identified
Connection matrix of graph parameter f
M ( f , k ) F F = f ( F1F2 )
1 2
k=2:
...
f(
...
)
f is a moment parameter
L-Szegedy
Û
f ( F ) = lim t ( F , Gn )
Û
f ( K1 ) = 1, f multiplicative
M ( f , k ) positive semidefinite
f is reflection positive
Gives inequalities between subgraph densities

extremal graph theory
Extremal graph theory as properties of t Î T
Turán’s Theorem for triangles:
t ( )  t ( )(2t ( )  1)
Kruskal-Katona Theorem for triangles:
t( )  t( )
3/ 2
Graham-Chung-Wilson Theorem about quasirandom graphs:
t( )  p 
|E ( F )|


F
t
(
F
)

p
4
t( )  p 
Proof of Kruskal-Katona
k=2
t(
)2  t( )t( )  t( )t( )  t( )3
Moments
1-variable functions
t (k , f ) :=
ò
f k ( x) dx
[0,1]
2-variable functions
t ( F , W ) :=
ò
[0,1]V ( F )
Õ
W ( xi , x j ) dx
ij Î E ( F )
ErdősLSpencer
These are independent
quantities.
These are independent
quantities.
Moments determine the
function up to measure
preserving transformation.
Moments determine the
Borgsfunction up to measure
Chayespreserving transformation. L
Moment sequences
are characterized by
semidefiniteness
Moment graph parameters
Lare characterized by
Szegedy
semidefiniteness
Moment sequences are
interesting
Moment graph parameters
are interesting
t ( F , G ) = t ( F ,WG ) =
P(random map V ( F ) ® V (G ) preserves edges)
n|V ( F )|t ( F ,WKn ) = #(proper n-colorings of F )
partition functions,
homomorphism functions,...
2|E (G )| t ( F , cos(2p ( x - y ))) = # eulerian orientations of F
L-Szegedy
The following are cryptomorphic:
functions in W0 modulo measure preserving transformations
reflection positive and multiplicative graph parameters f
with f(K1)=1
random graph models G(n) that are
- label-independent
- hereditary
- independent on disjoint subsets
countable random graphs G that are
- label-independent
- independent on disjoint subsets
The structure of W0
Rectangle norm:
W
X
:= sup
S ,T
ò W ( x, y) dx dy
S´ T
Rectangle distance:
dX (W1 ,W2 ) :=
inf
j ,y :[0,1]® [0,1]
measure preserving
dX (G1 , G2 ) := dX (WG1 , WG2 )
W1j - W2y
dX (W1 , W2 ) = 0 Û
" F t ( F ,W1 ) = t ( F ,W2 )
(
WX := W0 /d = 0 , dX
X
)
Weak Regularity Lemma:
Frieze-Kannan
1/ e2
" W Î WX " e > 0 $ stepfunction U with £ 2
steps
such that dX (W , U ) £ e.
" W Î WX " e > 0 $ graph G with £ 2
such that dX (W , WG ) £ e.
WX is compact
L-Szegedy
2/ e2
nodes
For a sequence of graphs (Gn), the following are equivalent:
(i) t ( F , Gn ) is convergent " F
(iii) (WGn ) is convergent in WX
(iii) (Gn ) is Cauchy with respect to dX
random graphs
® 1/ 2
uniform attachment graphs ® 1- max( x, y )
preferential attachment graphs ® ln( x) ln( y )
Approximate uniqueness
Borgs-ChayesL-T.Sós-Vesztergombi
t ( F ,W1 ) - t ( F ,W2 ) £ E( F ) dX (W1,W2 )
" F with | V ( F ) |£ 2
Þ
4/ e2
- 8/ e2
t ( F , W1 ) - t ( F , W2 ) £ 2
dX (W1 , W2 ) £ e
If G1 and G2 are graphs on n nodes so that for all F with
| V ( F ) |£ 2
4/ e2
t ( F , G1 ) - t ( F , G2 ) £ 2
-
8/ e2
then G1 and G2 can be overlayed so that for all S , T Í V (G1 )
eG1 ( S , T ) - eG2 ( S , T ) £ en 2
Local testing for global properties
What to ask?
-Does it have an even number of nodes?
-How dense is it (average degree)?
-Is it connected?
For a graph parameter f, the following are equivalent:
(i) f can be computed by local tests
(ii) (Gn ) convergent Þ f (Gn ) convergent
(iii) f is unifomly continuous w.r.t dX
Borgs-ChayesL-T.Sós-Vesztergombi
Density of maximum cut is testable.
Key fact:
10
dX (G(n,W ),W ) <
log n
10
dX (G(n,WG ), G) <
log n