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Graph limit theory: an overview

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Graph limit theory: an overview

László Lovász

Eötvös Loránd University, Budapest

IAS, Princeton

Limit theories of discrete structures

rational numbers

trees

graphs

digraphs

hypergraphs

permutations

posets

abelian groups

metric spaces

Aldous, Elek-Tardos

Diaconis-Janson

Elek-Szegedy

Kohayakawa

Janson

Szegedy

Gromov Elek

Common elements in limit theories

sampling

sampling distance

limiting sample distributions

combined limiting sample

distributions

limit object

overlay distance

regularity lemma

applications

trees

graphs

digraphs

hypergraphs

permutations

posets

abelian groups

metric spaces

Limit theories for graphs

Dense graphs:

Borgs-Chayes-L-Sós-Vesztergombi

L-Szegedy

Inbetween: distances Bollobás-Riordan

regularity lemma Kohayakawa-Rödl, Scott

LaplacianChung

Bounded degree graphs:

Benjamini-Schramm, Elek

Left and right data

very large graph

counting colorations,

stable sets,

statistical physics,

maximum cut,

...

countingedges,

triangles,

...

spectra,

...

Dense graphs: convergence

distribution of k-samples

is convergent for everyk

t(F,G):Probability that random mapV(F)V(G)

preserves edges

(G1,G2,…) convergent:Ft(F,Gn) is convergent

Dense graphs: limit objects

W0 = {W: [0,1]2[0,1], symmetric, measurable}

"graphon"

GnW : F: t(F,Gn) t(F,W)

Graphs to graphons

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

G

AG

WG

Dense graphs: basic facts

For every convergent graph sequence(Gn)

there is a WW0 such thatGnW.

Is thistheonlyuseful

notion of convergence

of densegraphs?

Conversely, W(Gn) such thatGnW.

W is essentially unique

(up to measure-preserving transformation).

Bounded degree: convergence

Local : neighborhood sampling

Benjamini-Schramm

Global : metric space

Gromov

Local-global : Hatami-L-Szegedy

Right-convergence,…

Borgs-Chayes-Gamarnik

Graphings

Graphing: bounded degree graph G on [0,1] such that:

E(G) is a Borel set in [0,1]2

measure preserving:

0

A

B

1

degB(x)=2

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)

Graphings: examples

V(G) = circle

x-

x

E(G) = {chords with angle }

x+

Graphings: examples

V(G) = {rooted 2-colored grids}

E(G) = {shift the root}

Graphings: examples

x+

x-

x

x

x-

x+

bipartite?

disconnected?

Graphings and involution-invariant distributions

x: random point of [0,1]

Gx: connected component of G containing x

Gx is a random connected graph with bounded degree

This distribution is "invariant" under shifting the root.

Everyinvolution-invariantdistributioncan be representedby a graphing.

Elek

Graph limits and involution-invariant distributions

graphs,graphings,

orinv-invdistributions

(Gn) locally convergent: Cauchy in d

Gn G:d(Gn,G) 0 (n )

inv-inv distribution

Graph limits and involution-invariant distributions

Everylocallyconvergentsequence of bounded-degreegraphs has a limiting inv-invdistribution.

Benjamini-Schramm

Is everyinv-invdistributionthe limit of a locallyconvergentgraphsequence?

Aldous-Lyons

Local-global convergence

(Gn) locally-globally convergent: Cauchy in dk

Gn G:dk(Gn,G) 0 (n )

graphing

Local-global graph limits

Everylocally-globallyconvergentsequence of bounded-degreegraphs has a limit graphing.

Hatami-L-Szegedy

Convergence: examples

Gn: random 3-regular graph

Fn: random 3-regular bipartite graph

Hn: GnGn

Expander

graphs

Largegirthgraphs

Convergence: examples

Local limit:Gn, Fn, Hn rooted 3-regular treeT

Containsrecentresultthatindependence ratio

is convergent.

Bayati-Gamarnik-Tetali

Conjecture: (Gn), (Fn) and (Hn) are locally-globally

convergent.

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}

Local-global convergence: dense case

Everyconvergentsequence of graphs is Cauchy indk

L-Vesztergombi

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 ).

Regularity lemma

nxn grid is

(, 2/18)-homogeneous.

>0 >0 bounded-degGS E(G), |S|<|V(G)|,

st. allcomponents of G-S are (,)-homogeneous.

Angel-Szegedy, Elek-Lippner

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.

f() exists in the bounded degree case.

Alon

Extremal graph theory

It is undecidablewhether

holdsforeverygraphG.

Hatami-Norin

It is undecidablewhetherthereis a graphingwith almost allr-neighborhoodsin a givenfamilyF .

Csóka

Extremal graph theory: dense graphs

Kruskal-Katona

Razborov 2006

Fisher

Goodman

Bollobás

Mantel-Turán

Lovász-Simonovits

1

0

1/2

2/3

3/4

1

Extremal graph theory: D-regular

D3/8

Harangi

0

D2/6