Graph limits and graph homomorphisms

1. Graph limits and graph homomorphisms László Lovász Microsoft Research lovasz@microsoft.com

2. Why define limits of graph sequences? I. Very large graphs: • Internet -Social networks • Ecological systems • VLSI • Statistical physics • Brain Is there a good "small" approximation? Is there a good "continuous" approximation?

3. Minimize always >1/16, arbitrarily close for random graphs II. Real numbers minimum is not attained in rationals Minimize density of 4-cycles in a graph with edge-density 1/2 minimum is not attained among graphs

4. Limits of sequences of dense graphs: Borgs, Chayes, L, Sós, B.Szegedy, Vesztergombi Limits of sequences of graphs with bounded degree: Aldous, Benjamini-Schramm, Lyons, Elek

5. Limits of graph sequences Which sequences are convergent? Is there a limit object? Which parameters are “continuous at infinity”?

6. independent set coloring triangles Homomorphism: adjacency-preserving map

7. Probability that random map V(G)V(H) is a hom Weighted version:

8. hom(G, ) = # of independent sets in G Examples:

9. Example: random graphs with probability 1 Which graph sequences are convergent?

10. Example: Paley graphs p: prime 1 mod 4 Quasirandom graphs Thomason Chung – Graham – Wilson (Gn: n=1,2,...)is quasirandom:

11. "Counting lemma": (Gn) is convergent  Cauchy in the -metric. Distance of graphs

12. Given  >0 difference at most 1 with  k2exceptions for subsets X,Y of parts Vi,Vj # of edges between X and Y is pij|X||Y|  (n/k)2 Approximating by small graphs Szemerédi's Regularity Lemma 1974 The nodes of  graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities).

13. X Y

14. Given  >0 difference at most 1 for subset X of V, # of edges in X is Weak Regularity Lemma Frieze-Kannan 1989 The nodes of  graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities).

15. Corollary of the "weak" Regularity Lemma:

16. (G1, G2,...) convergent  Cauchy in the -metric. Limits of graph sequences Which sequences are convergent? Is there a limit object?

17. 1/2 A random graph with 100 nodes and with 2500 edges

18. A randomly grown uniform attachment graph with 100 nodes born at random times and with 2500 edges

19. A randomly grown preferential attachment graph with 100 fixed nodes and with 5,000 (multiple) edges

20. A randomly grown preferential attachment graph with 100 fixed nodes (ordered by degrees) and with 5,000 edges

21. The limit object as a function

22. Associated function WG: 0 1 0 1 1 0 1 1 Adjacency matrix of graph G: 0 1 0 1 1 1 1 0 Example 2: t(F,W)= 2-|E(F)| # of eulerian orientations of F Example 1:

23. Distance of functions

24. Restatement of the "Weak" Regularity Lemma:

25. For every convergent graph sequence(Gn) there is a such that Szemerédi Lemma Conversely, W(Gn) such that Summary of main results Wis essentially unique (up to measure-preserving transform).

26. (normalized) (multiplicative) The limit object as a graph parameter is a graph parameter "connection matrices" are positive semidefinite (reflection positive)

27. ... ... k=2:

28. Gives inequalities between subgraph densities  extremal graph theory

29. The limit object as a random graph model W-random graphs:

30. ergodic invariant measures on The following are cryptomorphic: functions in W0 modulo measure preserving transformations normalized, multiplicative and reflection positive graph parameters random graph models G(n) that are hereditary and independent on disjoint subsets

31. What to ask? -Does it have an even number of nodes? -How dense is it (average degree)? -Is it connected? Local testing for global properties

32. Sk: random set of k nodes f is testable  [(Gn) convergent  f(Gn) convergent] The density of the largest cut can be estimated by local tests. Goldreich - Goldwasser - Ron f is testable:

33. max cut