Graph limit theory: an overview László Lovász Eötvös Loránd University, Budapest IAS, Princeton June 20111

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 Elek 2

June 2011 Common elements in limit theories sampling sampling distance limiting sample distributions combined limiting sample distributions limit object overlay distance regularity lemma applications 3 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 very large graph counting edges, triangles,... spectra,... counting colorations, stable sets, statistical physics, maximum cut,... 5

June 2011 distribution of k-samples is convergent for every k t(F,G): Probability that random map V(F)  V(G) preserves edges (G 1,G 2,…) convergent:  F t(F,G n ) is convergent Dense graphs: convergence 6

June 2011 W 0 = { W: [0,1] 2  [0,1], symmetric, measurable } G n  W :  F: t(F,G n )  t(F,W) "graphon" Dense graphs: limit objects 7

G AGAG WGWG Graphs to graphons May 20128

June 2011 Dense graphs: basic facts For every convergent graph sequence (G n ) there is a W  W 0 such that G n  W. 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? 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: 01 deg B (x)=2 AB 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=G 1  …  G k measure preserving involutions (k  2D-1) 12

June 2011 Graphings: examples E(G) = {chords with angle  } x x-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 x x-x- x+x+ x x-x- x+x+ bipartite?disconnected? 15

June 2011 Graphings and involution-invariant distributions G x is a random connected graph with bounded degree x: random point of [0,1] G x : 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 graphs, graphings, or inv-inv distributions (G n ) locally convergent: Cauchy in d  G n  G: d  (G n,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 (G n ) locally-globally convergent: Cauchy in d  k G n  G: d k (G n,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 G n : random 3-regular graph F n : random 3-regular bipartite graph H n : G n  G n Large girth graphs Expander graphs 21

June 2011 Convergence: examples Local limit: G n, F n, H n  rooted 3-regular treeT 22 Conjecture: (G n ), (F n ) and (H n ) are locally-globally convergent. Contains recent result that independence ratio is convergent. Bayati-Gamarnik-Tetali

June 2011 Convergence: examples Local-global limit: G n, F n, H n tend to different graphings Conjecture: G n  T{0,1}, where V(T) = {rooted 2-colored trees} E(G) = {shift the root} 23

June 2011 Local-global convergence: dense case Every convergent sequence of graphs is Cauchy in d k L-Vesztergombi 24

June 2011 Regularity lemma Given an arbitrarily large graph G and an  >0, decompose G into f(  ) "homogeneous" parts. ( ,  )-homogeneous graph:  S  E(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. f(  ) exists in the bounded degree case. Alon 27

June 2011 Extremal graph theory It is undecidable whether holds for every graph G. 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 1/22/33/4 Razborov 2006 Mantel-Turán Goodman Fisher Lovász-Simonovits June Extremal graph theory: dense graphs

D 3 /8 D 2 /60 June Extremal graph theory: D-regular Harangi