1. July 20091 The Mathematical Challenge of Large Networks László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu

2. July 20092 What properties to study? - Does it have an even number of nodes? - How dense is it (average degree)? - Is it connected? - What are the connected components? Very large graphs Questions

3. July 20093 Very large graphs Framework How to obtain information about them? - Graph is HUGE. - Not known explicitly, not even number of nodes.

4. July 20094 Very large graphs Dense framework How to obtain information about them? Dense case:  cn 2 edges. - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes.

5. July 20095 Very large graphs Sparse framework How to obtain information about them? Sparse case: Bounded degree (  d). - We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth.

6. July 20096 Very large graphs Alternatives How to obtain information about them? - Observing global process locally, for some time. - Observing global parameters (statistical physics).

7. July 20097 Very large graphs Models How to model them? Erdős-Rényi random graphs Albert-Barabási graphs Many other randomly growing models

8. July 20098 Very large graphs Approximations How to approximate them? - By smaller graphs Szemerédi partitions (regularity lemmas) - By larger graphs Graph limits

9. July 20099 Very large dense graphs Distance How to measure their distance? (a) (b) cut distance

10. July 200910 Very large dense graphs Distance How to measure their distance? (c) blow up nodes, or fractional overlay

11. July 200911 Very large dense graphs Distance Examples: 1/2

12. July 200912 Very large dense graphs Sampling Lemmas With large probability, Borgs-Chayes-Lovász-Sós-Vesztergombi With large probability, Alon-Fernandez de la Vega-Kannan-Karpinski+

13. July 200913 Approximating by smaller Regularity Lemmas Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon – Fisher - Krivelevich - M. Szegedy 2000

14. July 200914 Approximating by smaller Regularity Lemmas The nodes of  graph can be partitioned into a bounded number of essentially equal parts so that almost all bipartite graphs between 2 parts are essentially random (with different densities). with εk 2 exceptions given ε>0, # of parts k is between 1/ ε and f( ε ) difference at most 1 for subsets X,Y of the two parts, # of edges between X and Y is p|X||Y|  ε n 2

15. July 200915 Approximating by smaller Regularity Lemmas “Weak” Regularity Lemma (Frieze-Kannan): p ij : density between S i and S j G P : complete graph on V(G) with edge weights p ij

16. July 200916 Approximating by smaller Regularity Lemmas “Weak" Regularity Lemma (Frieze-Kannan):

17. July 200917 “Weak” Regularity Lemma Similarity distance Fact 1. This is a metric, computable by sampling Fact 2. Weak Szemerédi partition  partition most nodes into sets with small diameter LL – B. Szegedy

18. July 200918 “Weak” Regularity Lemma Algorithm - Select random nodes v 1, v 2,... - Put v i in U iff d 2 (v i,u)> ε for all u  U. - Begin with U= . - Stop if for more than 1/ε 2 trials, U did not grow. Algorithm to construct representatives of classes: size bounded by f(ε)

19. July 200919 “Weak” Regularity Lemma Algorithm Let U={u 1,...,u k }. Put a node v in V i iff i is the first index with d 2 (u i,v)  ε. Algorithm to decide in which class does v belong:

20. July 200920 Max Cut in huge graphs Sampling? cut with many edges

21. July 200921 Approximating by larger Convergent graph sequences (i) and (ii) are equivalent. (ii) (G 1, G 2,...) convergent: Cauchy in the -metric. (i) distribution of k-samples is convergent for all k t(F,G): Probability that random mapV(F)  V(G) preserves edges

22. July 200922 Approximating by larger Convergent graph sequences "Counting lemma": “Inverse counting lemma": if for all graphs F with k nodes, then (i) and (ii) are equivalent.

23. A random graph with 100 nodes and with 2500 edges 1/2 July 200923

24. Rearranging the rows and columns July 200924

25. Approximation by small: Szemerédi's Regularity Lemma July 200925

26. Approximation by small: Szemerédi's Regularity Lemma Nodes can be so labeled essentially random July 200926

27. A randomly grown uniform attachment graph with 200 nodes Approximation by infinite July 200927

28. December 200828 Approximating by larger Limit objects (graphon)

29. December 200829 Approximating by larger Limit objects Distance of functions

30. December 200830 Approximating by larger Limit objects Converging to a function (i) and (ii) are equivalent.

31. December 200831 Approximating by larger Limit objects LL – B. Szegedy For every convergent graph sequence (G n ) there is a such that. Conversely,  W  (G n ) such that W is essentially unique (up to measure-preserving transform). Borgs – Chayes - LL

32. Extension to sparse graphs? July 200932 Convergence of graph sequences Limit objects Left and right convergence DenseBounded degree Erdős-L-Spencer 1979 Borgs-Chayes-L-Sós -Vesztergombi 2006 Aldous 1989 Benjamini-Schramm 2001 (graphons) L-B.Szegedy 2006 (graphings) Elek 2007 Benjamini-Schramm 2001 Borgs-Chayes-L-Sós -Vesztergombi 2010? Borgs-Chayes-Kahn-L 2010?

33. Extension to sparse graphs? July 200933 Distance Property testing DenseBounded degree Goldreich-Goldwasser -Ron 1998 Goldreich-Ron 1997 Arora-Karger-Karpinski 1995 Borgs-Chayes-L-Sós -Vesztergombi 2008 ? Regularity Lemma Szemerédi, Frieze-Kannan, Alon-Fischer-Krivelevich -Szegedy, Tao, L-Szegedy,… ?