1. Random graphs and limits of graph sequences László Lovász Microsoft Research lovasz@microsoft.com

2. W -random graphs

3. Adjacency matrix of weighted graph G, viewed as a function in W 0 : W G -random graphs  generalized random graphs with model G

4. density of F in W

5. Convergent graph sequences (G n ) is convergent: Examples: Paley graphs (quasirandom) half-graphs closest neighbor graphs...

6. Does a convergent graph sequence have a limit? For every convergent (G n ) there is a function W  W 0 such that B.Szegedy-L GnWGnW a.s.

7. Uniqueness of the limit Borgs-Chayes-L WWW W WW WWW W W W W W

8. A random graph with 100 nodes and 2500 edges 1/2 Quasirandom  converges to 1/2

9. Growing uniform attachment graph If there are n nodes - with prob c/n, a new node is added, - with prob (n-c)/n, a new edge is added.

10. A growing uniform attachment graph with 200 nodes and 10000 edges

11. Fixed preferential attachment graph Fix n nodes For m steps choose 2 random nodes independently with prob proportional to (deg+1) and connect them

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

13. A preferential attachment graph with 100 fixed nodes ordered by degrees and with 5,000 edges

14. Moments 1-variable functions 2-variable functions These are independent quantities. These are independent quantities. Erdős- L- Spencer Moments determine the function up to measure preserving transformation. Moment sequences are characterized by semidefiniteness Moments determine the function up to measure preserving transformation. Borgs- Chayes- L Moment graph parameters are characterized by semidefiniteness L- Szegedy Except for multiplicativity over disjoint union:

15. k -labeled graph: k nodes labeled 1,...,k Connection matrix of graph parameter f Connection matrices

16. k=2:...

17. f is a moment parameter L-Szegedy Gives inequalities between subgraph densities  extremal graph theory f is reflection positive

18. Kruskal-Katona Theorem for triangles: Turán’s Theorem for triangles: Graham-Chung-Wilson Theorem about quasirandom graphs: Extremal graph theory as properties of

19. k=2 Proof of Kruskal-Katona

20. Moments 1-variable functions 2-variable functions These are independent quantities. These are independent quantities. Erdős- L- Spencer Moments determine the function up to measure preserving transformation. Moment sequences are characterized by semidefiniteness Moments determine the function up to measure preserving transformation. Borgs- Chayes- L Moment graph parameters are characterized by semidefiniteness L- Szegedy Moment sequences are interesting Moment graph parameters are interesting

21. partition functions, homomorphism functions,... L-Szegedy

22. The following are cryptomorphic: functions in W 0 modulo measure preserving transformations reflection positive and multiplicative graph parameters f with f(K 1 )=1 random graph models G ( n ) that are - label-independent - hereditary - independent on disjoint subsets countable random graphs G that are - label-independent - independent on disjoint subsets

23. Rectangle norm: Rectangle distance: The structure of W 0

24. Weak Regularity Lemma: is compact L-Szegedy Frieze-Kannan

25. For a sequence of graphs (G n ), the following are equivalent: (i) (iii) uniform attachment graphs preferential attachment graphs random graphs

26. Approximate uniqueness Borgs-Chayes- L-T.Sós-Vesztergombi If G 1 and G 2 are graphs on n nodes so that for all F with then G 1 and G 2 can be overlayed so that for all

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

28. For a graph parameter f, the following are equivalent: (i) f can be computed by local tests (ii) (iii) f is unifomly continuous w.r.t Density of maximum cut is testable. Borgs-Chayes- L-T.Sós-Vesztergombi

29. Key fact: