1. Regularity partitions and the topology of graphons László Lovász Eötvös Loránd University, Budapest Joint work Balázs Szegedy August 20101

2. 2 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 p ij ). with ε k 2 exceptions given ε>0, # of parts k satisfies 1/ ε  k  f( ε ) difference at most 1 for subsets X,Y of the two parts, # of edges between X and Y is p ij |X||Y|  ε( n/k) 2 The Szemerédi Regularity Lemma

3. August 20103 Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon – Fisher - Krivelevich - M. Szegedy 2000 Tao 2005 L-Szegedy 2006 The Szemerédi Regularity Lemma

4. Low rank matrix approximation - Frieze-Kannan August 20104 The many facets of the Lemma Probability, information theory - Tao Approximation theory - L-Szegedy Compactness - L-Szegedy Dimensionality - L-Szegedy Measure theory - Bollobás-Nikiforov Sparse Regularity Lemma Gerke, Kohayakawa, Luczak, Rödl, Steger, Hypergraph Regularity Lemma Frankl, Gowers, Nagle, Rödl, Schacht Arithmetic Regularity Lemma Green, Tao Regularity Lemma and ultraproducts Elek, Szegedy

5. August 2010 Graphons 5 Could be:

6. 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 August 20106

7. 7 Cut distance of graphons cut norm cut distance measure preserving

8. August 2010 Regularity Lemma and cut distance 8 P : measurable partition of [0,1], Weak Regularity Lemma: is compact Strongest Regularity Lemma:

9. August 2010 Subgraph density 9 Probability that random map V(F)  V(G) is a hom

10. August 2010 Graphons as limit objects 10 Borgs, Chayes, L, Sós, Vesztergombi

11. For every convergent graph sequence (G n ) there is a graphon such that August 201011 Graphons as limit objects Conversely, for every graphon W there is a graph sequence (G n ) such that L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs- Chayes- L

12. A randomly grown uniform attachment graph with 200 nodes August 201012 Example: Randomly growing graphs

13. August 201013 Example: Generalized random graph Random with density 1/3 Random with density 2/3 Random with density 1/3

14. August 201014 Example: Borsuk graphon W(x,y)=1(|x-y|>2-d -2 ) G n  W  G n has weak regularity partition with O(  d ) classes  Neighborhoods in G n have VC-dimension d+1  W has a d-dimensional „underlying space”  G n does not contain F d+1 as an induced subgraph F3:F3: G n : induced subgraph on n random nodes S d with uniform distribution

15. August 201015 The topology of a graphon s t v w u Squaring the adjacency matrix

16. August 201016 The topology of a graphon Complete metric spaces A = Borel sets  has full support After a lot of cleaning Compact Not always compact pure graphon

17. August 201017 Example: Generalized random graphs again (J,r G ): discrete (J,r W )  (J,r WoW ): (J,r GoG ):(J,r WoW ): Gromov-Wasserstein convergence

18. August 201018 Example: Borsuk graphon again G n : induced subgraph on n random nodes G’ n : randomly delete half of the edges from G n

19. August 201019 average ε-net regular partition  S  J Voronoi cells of S form a partition with  partition P ={V 1,...,V k } of [0,1]  v i  V i with Regularity and dimensionality

20. August 201020 Theorem.  ε>0, the metric space (J,r WoW ) can be partitioned into a set of measure <ε, and sets with diameter <ε. Regularity and dimensionality

21. August 201021 Extremal graph theory and dimensionality F: bipartite graph with bipartition (U,V), G: graph W: graphon F bi-induced subgraph of G:  U’,V’  V(G), disjoint, subgraph formed by edges between U’ and V’ is isomorphic to F F bi-induced subgraph of W:

22. August 201022 Extremal graph theory and dimensionality Theorem. F is not a bi-induced subgraph of W  W is 0-1 valued, (J,r W ) is compact, and has finite packing dimension. Key fact: VC-dimension of neighborhoods is bounded

23. August 201023 Extremal graph theory and dimensionality Corollary. P: hereditary bigraph property not containing all bigraphs. (J,W): pure graphon in its closure  W is 0-1 valued, (J,r W ) is compact and has bounded dimension.

24. August 201024 Extremal graph theory and dimensionality Corollary. P : hereditary graph property not containing all graphs, such that  W in its closure is 0-1 valued, (J,W): pure graphon in its closure  (J,r W ) is compact and has bounded dimension.

25. August 201025 Example P : triangle-free

26. Corollary. F is not a bi-induced subgraph of G   >0, G has a weak regularity partition with error  with at most classes. August 201026 Extremal graph theory, dimensionality and regularity