The Mathematical Challenge of Large Networks

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

2. Very large graphs Questions • 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?

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

4. Very large graphs Dense framework • How to obtain information about them? • Dense case: cn2 edges. • - We can sample a uniform random node a bounded number of times, and see edges • between sampled nodes.

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

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

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

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

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

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

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

13. Approximating by smaller Regularity Lemmas Original Regularity LemmaSzemerédi 1976 “Weak” Regularity LemmaFrieze-Kannan 1999 “Strong” Regularity LemmaAlon – Fisher - Krivelevich - M. Szegedy 2000

14. Approximating by smaller Regularity Lemmas given ε>0, # of partsk is between 1/ ε and f(ε) difference at most 1 with εk2 exceptions for subsetsX,Y of the two parts, # of edges betweenX andY is p|X||Y| εn2 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).

15. Approximating by smaller Regularity Lemmas “Weak” Regularity Lemma (Frieze-Kannan): pij: density between Si and Sj GP: complete graph on V(G) with edge weights pij

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

17. “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. “Weak” Regularity Lemma Algorithm size bounded by f(ε) Algorithm to construct representatives of classes: - Begin with U=. - Select random nodes v1, v2, ... - Put vi in U iff d2(vi,u)>ε for all uU. - Stop if for more than 1/ε2 trials, U did not grow.

19. “Weak” Regularity Lemma Algorithm Algorithm to decide in which class does v belong: Let U={u1,...,uk}. Put a node vinViiff i is the first index withd2(ui,v)ε.

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

21. Approximating by larger Convergent graph sequences (ii) (i) (G1,G2,...) convergent: Cauchy in the -metric. distribution of k-samples is convergent for all k t(F,G):Probability that random mapV(F)V(G) preservesedges (i) and (ii) are equivalent.

22. 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. 1/2 A random graph with 100 nodes and with 2500 edges

24. Rearranging the rows and columns

25. Approximation by small: Szemerédi's Regularity Lemma

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

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

28. Approximating by larger Limit objects (graphon)

29. Approximating by larger Limit objects Distance of functions

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

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

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

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