1. The mathematical challenge of large networks László Lovász Eötvös Loránd University, Budapest Joint work with Christian Borgs, Jennifer Chayes, Balázs Szegedy, Vera Sós and Katalin Vesztergombi May 20121

2. Minimize x 3 -6x over x  0 minimum is not attained in rationals Minimize 4-cycle density in graphs with edge-density 1/2 minimum is not attained among graphs always >1/16, arbitrarily close for random graphs Real numbers are useful Graph limits are useful May 20122 Graph limits: Why are they needed?

3. Two extremes: - dense (cn 2 edges) - bounded degree well developed good warm-up case How dense is the graph? May 20123 Inbetween??? Bollobas-Riordan Chung Conlon-Fox-Zhao less developed, more difficult most applications

4. May 20124 How is the graph given? - Graph is HUGE. - Not known explicitly, not even the number of nodes. Idealize: define minimum amount of info.

5. May 20125 Dense case:  cn 2 edges. - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. Bounded degree (  d) - We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth. How is the graph given? „Property testing”: Arora-Karger-Karpinski, Goldreich-Goldwasser-Ron, Rubinfeld-Sudan, Alon-Fischer-Krivelevich-Szegedy, Fischer, Frieze-Kannan, Alon-Shapira

6. May 20126 Lecture plan Want to construct completion of the set of graphs. - What is the distance of two graphs? - Which graph sequences are convergent? - How to represent the limit object? - How does this completion space look like? - How to approximate graphs? (Regularity Lemmas and sampling)

7. May 20127 Lecture plan Applications in (dense) extremal graph theory - Are extremal graph problems decidable? - Which graphs are extremal? - Local vs. global extrema - Is there always an extremal graph?

8. May 20128 Lecture plan Applications in property testing - Deterministic and non-deterministic sampling (P=NP) - Which properties are testable by sampling?

9. May 20129 Lecture plan The bounded degree case - Different limit objects: involution-invariant distributions, graphings - Local algorithms and distributed computing

10. 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 Pixel pictures May 201210

11. A random graph with 100 nodes and with 2500 edges May 201211 Pixel pictures

12. Rearranging rows and columns May 201212 Pixel pictures

13. May 201213 Pixel pictures A randomly grown uniform attachment graph on 200 nodes At step n: - a new node is born; - any two nodes are joined with probability 1/n Ignore multiplicity of edges

14. Approximation by small: Regularity Lemma Szemerédi 1975 May 201214

15. Nodes can be so ordered essentially random Approximation by small: Regularity Lemma May 201215

16. The nodes of any 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). with  k 2 exceptions for subsets X,Y of parts V i,V j # of edges between X and Y is p ij |X||Y| ±  ( n/k) 2 Given  >0 difference at most 1 Approximation by small: Regularity Lemma May 201216

17. May 201217 Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon – Fisher - Krivelevich - M. Szegedy 2000 Approximation by small: Regularity Lemma

18. May 201218 A randomly grown uniform attachment graph on 200 nodes Graph limits: Examples

19. Knowing the limit W knowing many properties (approximately). Graph limits: Why are they useful? triangle density  May 201219

20. May 201220 Limit objects: the math distribution of k-samples is convergent for all 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

21. May 201221 Limit objects: the math W 0 = { W: [0,1] 2  [0,1], symmetric, measurable } G n  W :  F: t(F,G n )  t(F,W) "graphon"

22. Randomly grown prefix attachment graph At step n: - a new node is born; - connects to a random previous node and all its predecessors May 201222 Limit objects: an example

23. A randomly grown prefix attachment graph with 200 nodes Is this graph sequence convergent at all? Yes, by computing subgraph densities! This tends to some shades of gray; is that the limit? No, by computing triangle densities! May 201223

24. A randomly grown prefix attachment graph with 200 nodes (ordered by degrees) This also tends to some shades of gray; is that the limit? No… May 201224 Limit objects: an example

25. The limit of randomly grown prefix attachment graphs May 201225 Limit objects: an example

26. The distance of two graphs May 201226 (a) (b) cut distance (c) blow up nodes finite definition by fractional overlay

27. May 201227 Examples: The distance of two graphs

28. May 201228 p ij : density between S i and S j G P : complete graph on V(G) with edge weights p ij “Weak” Regularity Lemma

29. May 201229 Frieze-Kannan “Weak” Regularity Lemma

30. May 201230 Counting lemma: Inverse counting lemma: If for all graphs F with k nodes, then Counting Lemmas

31. May 201231 Equivalence of convergence notions A graph sequence (G 1,G 2,...) is convergent iff it is Cauchy in  .

32. December 200832 The distance of two graphons

33. May 201233 The semimetric space ( W 0,   ) is compact. “Strong” Regularity Lemma

34. May 201234 Limit objects: the math 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.

35. May 201235 Limit objects: the math 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. Regularity Lemma + martingales L-Szegedy Exchangeable random variables Aldous; Diaconis-Janson Ultraproducts Elek-Szegedy

36. May 201236 Limit objects: the math 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. W-random graphs (sampling from a graphon)

37. May 201237 Limit objects: the math 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. Constructing canonical representation Borgs-Chayes-L Exchangeable random variables Kallenberg; Diaconis-Janson Inverse Counting Lemma, measure compactness Bollobas-Riordan

38. May 201238