graph limit theory: an overview lászló lovász eötvös loránd university, budapest ias,...
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Graph limit theory: an overview
László Lovász
Eötvös Loránd University, Budapest
IAS, Princeton
June 2011 1
June 2011
Limit theories of discrete structures
trees
graphs
digraphs
hypergraphs
permutations
posets
abelian groups
metric spaces
rational numbers
Aldous, Elek-Tardos
Diaconis-Janson
Elek-Szegedy
Kohayakawa
Janson
Szegedy
Gromov Elek2
June 2011
Common elements in limit theories
sampling
sampling distance
limiting sample distributions
combined limiting sample
distributions
limit object
overlay distance
regularity lemma
applications3
trees
graphs
digraphs
hypergraphs
permutations
posets
abelian groups
metric spaces
June 2011
Limit theories for graphs
Dense 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
4
June 2011
Left and right data
F HG® ®
very large graph
counting edges,triangles,...spectra,...
counting colorations,stable sets,statistical physics,maximum cut,...
5
June 2011
t(F,G): Probability that random map V(F)V(G) preserves edges
(G1,G2,…) convergent: F t(F,Gn) is convergent
Dense graphs: convergence
6
June 2011
W0 = {W: [0,1]2 [0,1], symmetric, measurable}
( ) ( )[0,1]
( ,( , ) )Î
= ÕòV F
i jij E F
W x x dxt F W
GnW : F: t(F,Gn) t(F,W)
"graphon"
Dense graphs: limit objects
7
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
Graphs to graphons
May 2012 8
June 2011
Dense graphs: basic facts
For every convergent graph sequence (Gn)
there is a WW0 such that GnW .
W is essentially unique
(up to measure-preserving transformation).
Conversely, W (Gn) such that GnW . Is this the only useful
notion of convergence
of dense graphs?
9
June 2011
Bounded degree: convergence
Local : neighborhood sampling
Benjamini-Schramm
Global : metric space
Gromov
Local-global : Hatami-L-Szegedy
Right-convergence,…
Borgs-Chayes-Gamarnik
10
June 2011
Graphings
Graphing: bounded degree graph G on [0,1] such that:
E(G) is a Borel set in [0,1]2
measure preserving: deg ( ) deg ( )B A
A B
x dx x dx=ò ò
0 1
degB(x)=2
A B
11
June 2011
Graphings
Every Borel subgraph of a graphing is a graphing.
Every graph you ever want to construct from a
graphing is a graphing
D=1: graphing measure preserving involution
G is a graphing G=G1… Gk measure
preserving involutions (k2D-1)
12
June 2011
Graphings: examples
E(G) = {chords with angle }x x-
x+
V(G) = circle
13
June 2011
Graphings: examples
V(G) = {rooted 2-colored grids}
E(G) = {shift the root}
14
June 2011
Graphings: examples
xx-
x+
x x-
x+
bipartite? disconnected?
15
June 2011
Graphings and involution-invariant distributions
Gx is a random connected graph with bounded degree
x: random point of [0,1]
Gx: connected component of G containing x
This distribution is "invariant" under shifting the root.
Every involution-invariant distribution can be represented by a graphing.
Elek
16
June 2011
Graph limits and involution-invariant distributions
total variation distance
of -neighborh od
,
s
( )
o
r
r
d G H =e
2 )( ,, () r r
r
d G H d G H-=å ee
graphs, graphings,or inv-inv
distributions
(Gn) locally convergent: Cauchy in d
Gn G: d (Gn,G) 0 (n )
inv-inv distribution
17
June 2011
Graph limits and involution-invariant distributions
Every 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
18
June 2011
Local-global convergence
min : -coloring * of
-coloring * of
st. sampling ( *, *)
( ,
and vice versa
)k c k G G
k H H
d
d H
G H
G
c
"
$
=
<e
e
(Gn) locally-globally convergent: Cauchy in dk
Gn G: dk(Gn,G) 0 (n )
graphing
19
June 2011
Local-global graph limits
Every locally-globally convergent sequence of bounded-degree graphs has
a limit graphing.
Hatami-L-Szegedy
20
June 2011
Convergence: examples
Gn: random 3-regular graph
Fn: random 3-regular bipartite graph
Hn: GnGn
Large girth graphs
Expandergraphs
21
June 2011
Convergence: examples
Local limit: Gn, Fn, Hn rooted 3-regular treeT
22
Conjecture: (Gn), (Fn) and (Hn) are locally-globally
convergent.
Contains recent result
that independence ratio
is convergent.
Bayati-Gamarnik-Tetali
June 2011
Convergence: examples
Local-global limit: Gn, Fn, Hn tend to different graphings
Conjecture: Gn T{0,1}, where
V(T) = {rooted 2-colored trees}
E(G) = {shift the root}
23
June 2011
Local-global convergence: dense case
min : -coloring * of
-coloring * of
st. sampling distance( *, *)
and vice versa
( , )k c k G G
k H H
G H c
d G H "
<
=
$
Every convergent sequence of graphs
is Cauchy in dk
L-Vesztergombi
24
June 2011
Regularity lemma
Given 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 ).
25
June 2011
Regularity lemma
nxn 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
26
June 2011
Regularity lemma
Given an arbitrarily large graph G and an >0,
find a graph H of size at most f() such that
G and H are -close in sampling distance.
Frieze-Kannan "Weak" Regularity Lemma
suffices in the dense case.24 /2( )f ee =
f() exists in the bounded degree case.
Alon
27
June 2011
Extremal graph theory
It is undecidable whether
holds for every graph G.
( , ) 0FF
a t F G ³å Hatami-Norin
It is undecidable whether there is a graphing with
almost all r-neighborhoods in a given family F .
Csóka
28
1
10
Kruskal-Katona
Bollobás
1/2 2/3 3/4
Razborov 2006
Mantel-Turán
Goodman
Fisher
Lovász-SimonovitsJune 2011 29
Extremal graph theory: dense graphs
D3/8
D2/60
June 2011 30
Extremal graph theory: D-regular
Harangi