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Regularity partitions and the topology of graphons

Regularity partitions and the topology of graphons. L á szl ó Lov á sz Eötvös Lor ánd University, Budapest. Joint work Bal á zs Szegedy. The Szemerédi Regularity Lemma. given ε >0, # of parts k satisfies 1 / ε  k  f ( ε ). difference at most 1. with ε k 2 exceptions.

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Regularity partitions and the topology of graphons

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  1. Regularity partitions and the topology of graphons László Lovász Eötvös Loránd University, Budapest Joint workBalázs Szegedy

  2. The Szemerédi Regularity Lemma given ε>0, # of parts k satisfies 1/ ε kf(ε) difference at most 1 with εk2 exceptions for subsets X,Y of the two parts, # of edges between X and Y is pij|X||Y| ε(n/k)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 pij).

  3. The Szemerédi Regularity Lemma Original Regularity LemmaSzemerédi 1976 “Weak” Regularity LemmaFrieze-Kannan 1999 “Strong” Regularity LemmaAlon – Fisher - Krivelevich - M. Szegedy 2000 Tao 2005 L-Szegedy 2006

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

  5. Graphons Could be:

  6. Pixel pictures 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 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 G AG WG

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

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

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

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

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

  12. Example: Randomly growing graphs A randomly grown uniform attachment graphwith 200 nodes

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

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

  15. The topology of a graphon w t u s v Squaringtheadjacencymatrix

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

  17. Example: Generalized random graphs again (J,rG): discrete (J,rW)  (J,rWoW): (J,rGoG): (J,rWoW): Gromov-Wasserstein convergence

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

  19. Regularity and dimensionality SJVoronoi cells of S form a partition with  partition P={V1,...,Vk}of [0,1]  viVi with average ε-net regular partition

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

  21. Extremal graph theory and dimensionality F: bipartitegraphwithbipartition (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. Extremal graph theory and dimensionality Theorem. F is not a bi-inducedsubgraph of W  W is 0-1 valued, (J,rW) is compact, and has finitepackingdimension. Key fact: VC-dimension of neighborhoods is bounded

  23. 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,rW) is compact and has bounded dimension.

  24. 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,rW)is compact and has bounded dimension.

  25. Example P: triangle-free

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

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