Graph limit theory: an overview. L á szl ó Lov á sz Eötvös Lor ánd 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
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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