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

2. 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

3. 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

4. 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

5. June 2011 Left and right data very large graph counting edges, triangles,... spectra,... counting colorations, stable sets, statistical physics, maximum cut,... 5

6. 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

7. 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

8. 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 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 AGAG WGWG Graphs to graphons May 20128

9. 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

10. 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

11. 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

12. 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

13. June 2011 Graphings: examples E(G) = {chords with angle  } x x-x- x+x+ V(G) = circle 13

14. June 2011 Graphings: examples V(G) = {rooted 2-colored grids} E(G) = {shift the root} 14

15. June 2011 Graphings: examples x x-x- x+x+ x x-x- x+x+ bipartite?disconnected? 15

16. 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

17. 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

18. 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

19. 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

20. June 2011 Local-global graph limits Every locally-globally convergent sequence of bounded-degree graphs has a limit graphing. Hatami-L-Szegedy 20

21. 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

22. 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

23. 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

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

25. 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

26. 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

27. 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

28. 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

29. 1 10 Kruskal-Katona Bollobás 1/22/33/4 Razborov 2006 Mantel-Turán Goodman Fisher Lovász-Simonovits June 201129 Extremal graph theory: dense graphs

30. D 3 /8 D 2 /60 June 201130 Extremal graph theory: D-regular Harangi