L á szl ó Lov á sz Eötvös Lor ánd University Budapest IAS Princeton June 2011 1 June 2011 Limit theories of discrete structures trees graphs digraphs ID: 322555
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Slide1
Graph limit theory: an overview
László Lovász Eötvös Loránd University, BudapestIAS, Princeton
June 2011
1Slide2
June 2011
Limit theories of discrete structurestreesgraphsdigraphshypergraphspermutationsposets
abelian
groups
metric
spaces
rational numbers
Aldous, Elek-TardosDiaconis-JansonElek-SzegedyKohayakawaJansonSzegedyGromov Elek
2Slide3
June 2011
Common elements in limit theoriessamplingsampling distancelimiting sample
distributions
combined
limiting
sample
distributionslimit object
overlay distanceregularity lemmaapplications3treesgraphsdigraphshypergraphs
permutations
posets
abelian
groups
metric
spacesSlide4
June 2011
Limit theories for graphsDense graphs: Borgs-Chayes-L-Sós-Vesztergombi
L-Szegedy
Bounded
degree
graphs:
Benjamini-Schramm, Elek
Inbetween: distances Bollobás-Riordan regularity lemma Kohayakawa-Rödl, Scott Laplacian Chung
4Slide5
June 2011
Left and right datavery large graph
counting
edges
,
triangles,...
spectra,...
counting colorations,stable sets,statistical physics,maximum cut,...5Slide6
June 2011
distribution of k-samplesis convergent for every kt(F,G): Probability that random map
V(
F
)
V(G)
preserves edges(
G1,G2,…) convergent: F t(F,Gn) is convergent
Dense
graphs
:
convergence
6Slide7
June 2011
W0 = {W: [0,1]2 [0,1], symmetric, measurable}GnW :
F
:
t(F,
Gn) t(
F,W)"graphon"Dense graphs: limit objects7Slide8
G
0 0 1 0 0 1 1 0 0 0 1 0 0 1
0 0 1 0 1 0 1 0 0 0 0 0 1 0
1 1 0 1 0 1 1 1 1 0 1 0 1 1
0 0 1 0 1 0 1 0 1 0 1 1 0 0
0 1 0 1 0 1 1 0 0 0 1 0 0 1
1 0 1 0 1 0 1 1 0 1 1 1 0 1
1 1 1 1 1 1 0 1 0 1 1 1 1 0
0 0 1 0 0 1 1 0 1 0 1 0 1 1
0 0 1 1 0 0 0 1 1 1 0 1 0 0
0 0 0 0 0 1 1 0 1 0 1 0 1 0
1 0 1 1 1 1 1 1 0 1 0 1 1 1
0 0 0 1 0 1 1 0 1 0 1 0 1 0
0 1 1 0 0 0 1 1 0 1 1 1 0 1
1 0 1 0 1 1 0 1 0 0 1 0 1 0
A
G
W
G
Graphs
to
graphons
May 2012
8Slide9
June 2011
Dense graphs: basic factsFor every convergent graph sequence (Gn)there is a W
W
0
such that
GnW
.W is essentially unique (up to measure-preserving transformation).Conversely,
W
(
G
n
)
such
that
G
n
W
.
Is
this
the
only
useful
notion
of
convergence
of
dense
graphs
?
9Slide10
June 2011
Bounded degree: convergenceLocal : neighborhood sampling Benjamini-SchrammGlobal : metric
space
Gromov
Local-global
: Hatami-L-Szegedy
Right-convergence,… Borgs-Chayes-Gamarnik10Slide11
June 2011
GraphingsGraphing: bounded degree graph G on [0,1] such that:
E
(
G
) is a
Borel set
in [0,1]2
measure preserving: 01
deg
B
(x)=2
A
B
11Slide12
June 2011
GraphingsEvery Borel subgraph of a graphing is a graphing.Every
graph
you
ever
want to
construct from a graphing
is a graphingD=1: graphing measure preserving involutionG is a graphing
G
=
G
1
…
G
k
measure
preserving
involutions
(
k
2
D
-1)
12Slide13
June 2011
Graphings: examplesE(G)
= {chords
with
angle }
x
x-x+
V
(
G
)
=
circle
13Slide14
June 2011
Graphings: examples
V
(
G
)
= {
rooted
2-colored
grids
}
E
(
G
)
= {shift
the
root
}
14Slide15
June 2011
Graphings: examplesx
x
-
x
+
xx-x+
bipartite
?
disconnected
?
15Slide16
June 2011
Graphings and involution-invariant distributionsGx is a random connected graph with bounded degree
x
:
random
point
of [0,1]
Gx: connected
component of G containing xThis distribution is "invariant" under shifting the
root
.
Every
involution-invariant
distribution
can
be
represented
by
a
graphing
.
Elek
16Slide17
June 2011
Graph limits and involution-invariant distributionsgraphs, graphings,or
inv-inv
distributions
(
Gn
) locally convergent:
Cauchy in dGn G: d (Gn
,
G
)
0 (
n
)
inv-inv
distribution
17Slide18
June 2011
Graph limits and involution-invariant distributionsEvery locally convergent sequence
of bounded-degree
graphs
has a limiting
inv-inv distribution
.Benjamini-Schramm
Is every inv-inv distribution the limit of a locally convergent graph sequence?
Aldous-Lyons
18Slide19
June 2011
Local-global convergence(Gn) locally-globally convergent: Cauchy in
d
k
G
n
G:
dk(Gn,G) 0 (n )graphing19Slide20
June 2011
Local-global graph limitsEvery locally-globally convergent sequence of bounded-degree
graphs
has a limit
graphing
.Hatami-L-Szegedy
20Slide21
June 2011
Convergence: examplesGn: random 3-regular graphFn: random 3-regular bipartite graphH
n:
G
n
Gn
Large
girth graphs
Expander
graphs
21Slide22
June 2011
Convergence: examplesLocal limit: Gn, Fn, Hn rooted
3-regular tree
T
22
Conjecture
:
(
G
n
)
,
(
F
n
) and
(
H
n
)
are
locally-globally
convergent
.
Contains
recent
result
that
independence
ratio
is
convergent
.
Bayati-Gamarnik-TetaliSlide23
June 2011
Convergence: examplesLocal-global limit: Gn, Fn, Hn tend
to
different
graphings
Conjecture
:
G
n
T
{0,1},
where
V
(
T
)
= {
rooted
2-colored
trees
}
E
(
G
)
= {shift
the
root
}
23Slide24
June 2011
Local-global convergence: dense caseEvery convergent
sequence
of
graphs
is Cauchy
in d
kL-Vesztergombi
24Slide25
June 2011
Regularity lemmaGiven an arbitrarily large graph G and an >0,decompose
G
into
f
()
"homogeneous" parts.
(,)-homogeneous graph: SE(G), |S|<|V(
G
)|,
all
connected
components
of
G
-
S
with
> |
V
(
G
)|
nodes
have
the
same
neighborhood
distribution
(
up
to
).
25Slide26
June 2011
Regularity
lemma
n
x
n
grid is (,
2/18)-homogeneous.>0 >0 bounded-deg G S
E
(
G
), |
S
|<|
V
(
G
)|,
st
.
all
components
of
G
-
S
are
(,)
-homogeneous
.
Angel-Szegedy
,
Elek-Lippner
26Slide27
June 2011
Regularity lemmaGiven an arbitrarily large graph G and an >0,find
a graph
H
of
size
at most f() such
thatG and H are -close in sampling distance.
Frieze-Kannan
"
Weak
"
Regularity
Lemma
suffices
in
the
dense
case
.
f
(
)
exists
in
the
bounded
degree
case
.
Alon
27Slide28
June 2011
Extremal graph theory It is undecidable whether
holds
for
every graph
G.
Hatami-NorinIt is undecidable whether there is a graphing
with
almost
all
r
-neighborhoods
in
a
given
family
F
.
Csóka
28Slide29
1
1
0
Kruskal-Katona
Bollob
ás
1/2
2/3
3/4
Razborov 2006
Mantel-Tur
án
Goodman
Fisher
Lov
ász-Simonovits
June 2011
29
Extremal
graph
theory
:
dense
graphs
Slide30
D
3/8D2/60June 201130Extremal
graph
theory
:
D-regular
Harangi