1. October 20091 Large networks: a new language for science László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu

2. October 20092 Joint work with Christian Borgs Jennifer Chayes Vera T. Sós Balázs Szegedy Katalin Vesztergombi

3. @Stephen Coast Very large graphs - internet October 20093

4. 4 Very large graphs Examples - internet - social networks How fast do inventions news diseases religions spread?

5. October 20095 Very large graphs Examples - internet - social networks - ecological networks - chip design Institut f. Diskrete Mathematik Universität Bonn

6. October 20096 Very large graphs Examples - internet - social networks - ecological networks - chip design - statistical physics Ising model: Phase transition depends on the graph.

7. October 20097 Very large graphs Examples - internet - social networks - ecological networks - chip design - statistical physics - human brain What can mathematics and theoretical computer science contribute? The symbiosis of physics and analysis computer science and discrete math biology and …

8. October 20098 - What properties to study? - How to obtain information about them? - How to model them? - How to approximate them? - How to measure their distance? - How to return the result? Very large graphs Problems:

9. October 20099 - Does it have an even number of nodes? - How dense is it (average degree)? - Is it connected? - What are the connected components? What properties to study?

10. October 200910 How to obtain information about them? - Graph is HUGE. - Not known explicitly, not even number of nodes. „Property testing”: Arora-Karger-Karpinski Goldreich-Goldwasser-Ron Rubinfeld-Sudan Fischer Frieze-Kannan

11. October 200911 - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. Works in the dense case only (  cn 2 edges) How to obtain information about them?

12. October 200912 - We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth. Works in the sparse case: Bounded degree (  d). How to obtain information about them?

13. October 200913 - Observing global process locally, for some time. - Observing global parameters (statistical physics). How to obtain information about them?

14. October 200914 How to model them? Erdős-Rényi random graphs - n nodes - connect any two with probability p, independently

15. October 200915 The distribution of the degrees of a random graph average degree=10 How to model them?

16. October 200916 Very large graphs Models average degree=10 The distribution of the degrees of a real life graph

17. October 200917 Albert - Barabási graphs: - at every step a new node is born; - selects d old nodes with probability proportional to the degree, and connects there. Many other growing models How to model them?

18. October 200918 How to approximate them? - By smaller graphs Szemerédi partitions (regularity lemmas) - By larger graphs Graph limits

19. October 200919 How to measure their distance? (a) (b) cut distance (c) blow up nodes, or fractional overlay

20. October 200920 Examples: How to measure their distance?

21. October 200921 Sampling Lemmas With large probability, Alon-Fernandez de la Vega-Kannan-Karpinski+

22. October 200922 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

23. October 200923 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 ε k 2 exceptions given ε>0, # of parts 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

24. October 200924 Approximating by smaller Regularity Lemmas “Weak" Regularity Lemma (Frieze-Kannan):

25. October 200925 “Weak” Regularity Lemma Geometric version Fact 1. This is a metric. Similarity metric on V(G): s t v w u

26. October 200926 “Weak” Regularity Lemma Geometric version Fact 2. Weak Szemerédi partition  partition most nodes into sets with small diameter

27. October 200927 “Weak” Regularity Lemma Algorithm - Select random nodes v 1, v 2,... - Add v i to 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)

28. October 200928 “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:

29. October 200929 Max Cut in huge graphs Algorithm - Construct U as for the weak Szemerédi partition - Compute p ij = density between classes V i and V j (use sampling) - Compute max cut (U 1,U 2 ) in complete graph on U with edge-weights p ij Algorithm to construct representation of cut: (Different algorithm implicit by Frieze-Kannan.)

30. October 200930 Approximating by larger Convergent graph sequences Probability that random map V(F)  V(G) is a hom (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

31. October 200931 Approximating by larger Convergent graph sequences with probability 1 Example: random graphs

32. October 200932 Approximating by larger Limit objects 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

33. October 200933 Approximating by larger Limit objects A random graph with 100 nodes and with 2500 edges 1/2

34. October 200934 Approximating by larger Limit objects Rearranging the rows and columns

35. October 200935 Approximating by larger Limit objects A random graph with 100 nodes and with 2500 edges 1/2 (no matter how you reorder the nodes)

36. October 200936 Approximating by larger Limit objects A randomly grown uniform attachment graph with 200 nodes

37. October 200937 Approximating by larger Limit objects Limit object can be represented by a (symmetric, measurable) function W: [0,1] 2  [0,1] Δ-density

38. What this theory misses… October 200938 Real life graphs are mostly sparse. Real life networks have action on them - flows - random walks - information spreading Real life networks change in time Topics for many PhD’s for the next 25 years… Developing theory Not much for large graphs