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Around the Regularity Lemma

Around the Regularity Lemma. L á szl ó Lov á sz, Bal á zs szegedy Microsoft Research IAS Princeton. 1. Szemer édi’s Regularity Lemma. 2. The Regularity Lemma in Hilbert space. 3. The Regularity Lemma as compactness. 4. The Regularity Lemma as dimensionality. Given  >0.

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Around the Regularity Lemma

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  1. Around the Regularity Lemma László Lovász, Balázs szegedy Microsoft Research IAS Princeton

  2. 1. Szemerédi’s Regularity Lemma 2. The Regularity Lemma in Hilbert space 3. The Regularity Lemma as compactness 4. The Regularity Lemma as dimensionality

  3. Given  >0 difference at most 1 with  k2exceptions for subsets X,Y of parts Vi,Vj # of edges between X and Y is pij|X||Y|  (n/k)2 Szemerédi partition with error  Szemerédi's Regularity Lemma 1974 The nodes of  graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities).

  4. X Y

  5. Given  >0 difference at most 1 for subset X of V, # of edges in X is Regularity Lemma Light Frieze-Kannan 1989 The nodes of  graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities).

  6. Strong Regularity Lemma Alon, Fischer, Krivelevich, Szegedy (0,1,...,k,...) k0>0 such that G we can change at most 0|V(G)|2 edges so that the resulting graph G' has an equipartition Q=(V1,...,Vk)(kk0) s.t. 1i j k, XVi, YVj,

  7. A lemma about Hilbert space L- B.Szegedy Corollary: approximation by stepfunction

  8. Strong lemma Weak lemma Adjacency matrix of G, viewed as a function Stepfunction approximation  Weak regularity lemma

  9. Rectangle norm: Rectangle distance:

  10. is compact L-Szegedy Weak Regularity Lemma:

  11. Erdős- L- Spencer These are independent quantities. Moments determine the function up to measure preserving transformation. Borgs- Chayes- L Except for multiplicativity over disjoint union: Moment graph parameters are characterized by semidefiniteness L- Szegedy Moments 2-variable functions 1-variable functions These are independent quantities. Moments determine the function up to measure preserving transformation. Moment sequences are characterized by semidefiniteness Moment sequences are interesting Moment graph parameters are interesting

  12. L-Szegedy partition functions, homomorphism functions,...

  13. If G1and G2are graphs on n nodes so that for all F with then G1and G2 can be overlayed so that for all Approximate uniqueness Borgs-Chayes- L-T.Sós-Vesztergombi

  14. Applications: - Limits of graph sequences - Graph parameter testing - Extremal graph theory

  15. A random graph with 100 nodes and with 2500 edges

  16. A randomly grown uniform attachment graph with 100 nodes born at random times and with 2500 edges

  17. A randomly grown preferential attachment graph with 100 fixed nodes and with 5,000 (multiple) edges

  18. A randomly grown preferential attachment graph with 100 fixed nodes (ordered by degrees) and with 5,000 edges

  19. random graphs uniform attachment graphs preferential attachment graphs For a sequence of graphs (Gn), the following are equivalent: (i) (iii) (iii)

  20. What to ask? -Does it have an even number of nodes? -How dense is it (average degree)? -Is it connected? Local testing for global properties

  21. (iii) f is unifomly continuous w.r.t For a graph parameter f, the following are equivalent: (i) f can be computed by local tests (ii) Density of maximum cut is testable.

  22. Kruskal-Katona Theorem for triangles: Turán’s Theorem for triangles: Graham-Chung-Wilson Theorem about quasirandom graphs: Extremal graph theory as properties of

  23. Connection matrix of graph parameter f Connection matrices k-labeled graph: k nodes labeled 1,...,k

  24. ... k=2: ...

  25. f is moment parameter Gives inequalities between subgraph densities L-Szegedy

  26. k=2 Proof of Kruskal-Katona

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