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Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
Presentation at IAS
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Presentation at IAS

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  • 1. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Limits of Structures: a Unifying Approach ˇ Jaroslav N EŠET RIL (joint work with Patrice O SSONA DE M ENDEZ) Charles University Praha, Czech Republic June 2012, I.A.S.
  • 2. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Dichotomies Sparse vs Dense Structure vs Random (existence) (counting)
  • 3. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Definitions Finite relational language (graphs, digraphs, k -colored graphs, etc.) FO: all first-order formulas X ⊆ FO: a fragment Definition A sequence G1 , . . . , Gn , . . . is X-convergent if, for every φ ∈ X, the sequence φ , G1 , . . . , φ , Gn , . . . is convergent, where |{(v1 , . . . , vp ) : Gn |= φ (v1 , . . . , vp )}| φ , Gn = |Gn |p (if φ = φ (x1 , . . . , xp ) is a formula with p free variables).
  • 4. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Special Fragments QF Quantifier free formulas FO0 Formulas with no free variables = sentences FOp Formulas with p free variables FOlocal Local formulas
  • 5. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Example 1: L-convergence F → φF = (xi ∼ xj ) ij ∈E (F ) Then hom(F , G) φF , G = = t (F , G). |G||F | Hence. . . G1 , . . . , Gn is L-convergent if and only if it is QF-convergent.
  • 6. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Example 2: Elementary-convergence For φ ∈ FO0 , we have  1 if G |= φ , φ,G = 0 otherwise. FO0 -convergence is called elementary convergence. ... ... 4× ... ... ... . . . . . . . . . . . . . . .
  • 7. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Local Formulas FOlocal : all formulas φ with ≥ 1 free variables such that ∃rφ such that for every graph G it holds G |= φ (x1 , . . . , xp ) ⇐⇒ G[Nrφ (x1 ) ∪ · · · ∪ Nrφ (xp )] |= φ (x1 , . . . , xp ). FOlocal : all local formulas with p free variables (p ≥ 1). p Proposition (JN,POM) A sequence G1 , . . . , Gn , . . . of graphs is FO-convergent if and only if it is both FOlocal -convergent and FO0 -convergent.
  • 8. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Proof Follows from Gaifman locality theorem: Theorem For every first-order formula φ (x1 , . . . , xp ) there exist an integer r such that φ is equivalent to a Boolean combination of r -local formulas ξ (xi1 , . . . , xis ) and sentences of the form ∃y1 . . . ∃ym dist(yi , yj ) > 2r ∧ ψ(yi ) 1≤i <j ≤m 1≤i ≤m where ψ is r -local.
  • 9. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Example 3: BS-convergence Graphs with maximum degree at most D G1 , . . . , Gn is BS-convergent if and only if it is FOlocal -convergent. 1 Theorem (JN,POM) A sequence G1 , . . . , Gn of graphs with maximum degree at most D is BS-convergent if and only if it is FOlocal -convergent. Corollary A sequence G1 , . . . , Gn of graphs with maximum degree at most D is FO-convergent if and only if it is both BS-convergent and elementarily convergent. Limit object: disjoint union of a graphing and a countable graph.
  • 10. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Example 3: BS-convergence Graphs with maximum degree at most D G1 , . . . , Gn is BS-convergent if and only if it is FOlocal -convergent. 1 Theorem (JN,POM) A sequence G1 , . . . , Gn of graphs with maximum degree at most D is BS-convergent if and only if it is FOlocal -convergent. Corollary A sequence G1 , . . . , Gn of graphs with maximum degree at most D is FO-convergent if and only if it is both BS-convergent and elementarily convergent. Limit object: disjoint union of a graphing and a countable graph.
  • 11. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Example 3: BS-convergence Graphs with maximum degree at most D G1 , . . . , Gn is BS-convergent if and only if it is FOlocal -convergent. 1 Theorem (JN,POM) A sequence G1 , . . . , Gn of graphs with maximum degree at most D is BS-convergent if and only if it is FOlocal -convergent. Corollary A sequence G1 , . . . , Gn of graphs with maximum degree at most D is FO-convergent if and only if it is both BS-convergent and elementarily convergent. Limit object: disjoint union of a graphing and a countable graph.
  • 12. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion General Case
  • 13. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Theories theory: set of sentences (no free variables); complete theory: consistent theory t such that ∀φ : φ ∈ t or ¬φ ∈ t; T: space of all complete theories; Tp : space of all complete theories in the language augmented by p symbols of constants. ultrametric on Tp : dist(t1 , t2 ) = 2− min{qrank(φ ): φ ∈t1 t2 } (for t1 = t2 ). Then Tp is compact Polish standard Borel space (Tp , Σp ).
  • 14. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion FOp -convergence Theorem (JN,POM) Let G1 , . . . , Gn , . . . be a FOp -convergent sequence. Then there exists a probability measure P on (Tp , Σp ) invariant by the action of Sp on constant symbols (subgroup of isometries of Tp ) s.t. ∀φ ∈ FOp : P ({t : φ (c1 , . . . , cp ) ∈ t }) = lim φ , Gn , n→∞ where φ (c1 , . . . , cp ) is obtained by replacing xi by constant ci .
  • 15. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Proof sketch Define |{(v1 , . . . , vp ) : Th(Gn , v1 , . . . , vp ) = t }| Pn (t ) = . |Gn |p Then Pn ⇒ P (by using finiteness of non-equivalent formulas with given quantifier rank).
  • 16. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Ultraproduct Construction Theorem (JN,POM) Let G1 , . . . , Gn , . . . be an FOp -convergent sequence and let G = ∏U Gn be the ultraproduct of the Gn and let µ be the Loeb limit measure on the vertex set V of G. Then for every φ ∈ FOp it holds ··· 1φ ([x 1 ], . . . , [x p ]) dµ([x 1 ]) . . . dµ([x p ]) = lim φ , Gi . U Proof. Elek-Szegedy via theories. Remark General construction but not explicit.
  • 17. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Connection From ultraproduct to Tp The mapping τ : V p → Tp defined by τ([x 1 ], . . . , [x p ]) = Th(G, [x 1 ], . . . , [x p ]) is measurable and the pushforward τ∗ (µ) of µ is P: for every Borel subset X of Tp it holds (τ∗ (µ))(X ) = µ(τ −1 (X )) = P (X ). From Tp to ultraproduct Common model to all the theories in the support of P (p + 1)-saturated model (ω -saturated model) ultraproduct.
  • 18. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Examples (p = 1) Paths P is concentrated on (the theory of) the union of the rooted line and 2 rays, Complete binary trees P is concentrated on a countable set (root at leaves→ 1/2, at neighbors of leaves → 1/4, etc.) De Bruijn sequences the support of P is uncountable. Remark If bounded degree, ≈ BS-convergence. Forests of bounded diameter quite complicated.
  • 19. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion p = 1, cont’d “Sketch representation” in T2 by a multigraph: natural (measurable) projection π : T2 → T1 ; vertices: (measurable) sets π −1 (t ) for t ∈ T1 ∩ Supp(P ); natural involutive isometry ι of T2 (exchange of symbols c1 and c2 ); edges: {t1 , t2 } such that (c1 ∼ c2 ) ∈ ti ∈ π −1 (ti ), and ι(t1 ) = t2 .
  • 20. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion p = 1, cont’d t1 T1 t2 T2 ι π −1(t1) π −1(t2)
  • 21. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion p = 1, cont’d t1 T1 t2 T2 ι π −1(t1) π −1(t2) Remarks Similar to construction by root shifting; Universal multigraph (fixed edge involution, standard embedding); Extends to p > 1.
  • 22. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion p = 1, example T1 −1 2 2−2 2−3 2−4 2−5 ...
  • 23. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Toward explicit limits. . .
  • 24. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Class Taxonomy Ω(n1+ ) bounded edges degree Bounded Ω(n2) ultra sparse expansion edges minor closed ∀τ, d(G τ ) < ∞ ∀τ, χ(G τ ) < ∞ Nowhere dense Somewhere dense ∀τ, ω(G τ ) < ∞ ∃τ, ω(G τ ) = ∞
  • 25. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion FO-limit of rooted forests of height t The case t = 2: still quite easy. S3 √ nS3 αn S3 S√ n √ Sαn α n S√n n1/3 Sn1/3
  • 26. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion FO-limit of rooted forests of height t The case t = 3: more difficult. For instance: Gn = T1+a1 ,3+a2 ,...,2n−1+an , (a1 ,...,an )∈{0,1}n where Tx1 ,...,xk = rooted tree with Sx1 , . . . , Sxk under the root. ... T
  • 27. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion FO-limit of rooted trees of height t Theorem (JN,POM) General form of a limit object ≈ Vertices: (t1 , α1 , t2 , α2 , . . . , tk , αk ) where ti ∈ T is a complete theory of a rooted tree of height ≤ t, F (ti +1 ) ⊆ ti , and ∑ dim αi ≤ 2t + 1; Edges: {(t1 , α1 , t2 , α2 , . . . , tk , αk ), (t1 , α1 , t2 , α2 , . . . , tk , αk , tk +1 , αk +1 )}. in Tt × [0 ; 1]2t +1 , where T is the space of complete theories of rooted trees with height at most t.
  • 28. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Bounded Expansion Classes Definition A class C has bounded expansion if there exists f : N → N such that for every graph H it holds: If a ≤ k -subdivision of H is a subgraph of G ∈ C then H ≤ f (k ) |H |. Examples Bounded degree, planar, classes excluding a topological minor, etc.
  • 29. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Density in Bounded Expansion Classes Theorem (JN,POM) Let C be a bounded expansion class. 1 Let φ ∈ FOp . Then log |{(v1 , . . . , vp ) ∈ Ap : G |= φ (v1 , . . . , vp )}| lim sup ∈ {−∞, 0, 1, . . . , p}. A⊆G∈C log |A| 2 If C is hereditary and F is a graph, then log(#F ⊆ G) lim sup ∈ {−∞, 0, 1, . . . , α(F )}. G∈C log |G|
  • 30. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Low tree-depth decomposition Color the vertices of G by N colors, consider the subgraphs GI induced by subsets I of ≤ p colors.
  • 31. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Low tree-depth decomposition Color the vertices of G by N colors, consider the subgraphs GI induced by subsets I of ≤ p colors.
  • 32. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Low tree-depth decomposition Definition A d-tree-depth decomposition of a graph G is a vertex coloring V (G) → [N ] such that for every subset I of ≤ d colors, the subgraph GI induced by the colors in I has tree-depth at most |I | (i.e. is a subgraph of the closure of a rooted forest of height |I |). The minimum possible N is denoted χd (G). Theorem (JN,POM) For every class of graphs C the following are equivalent: for every integer d, sup{χd (G) : G ∈ C } < ∞; the class C has bounded expansion.
  • 33. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Low tree-depth decomposition Definition A d-tree-depth decomposition of a graph G is a vertex coloring V (G) → [N ] such that for every subset I of ≤ d colors, the subgraph GI induced by the colors in I has tree-depth at most |I | (i.e. is a subgraph of the closure of a rooted forest of height |I |). The minimum possible N is denoted χd (G). Theorem (JN,POM) For every class of graphs C the following are equivalent: for every integer d, sup{χd (G) : G ∈ C } < ∞; the class C has bounded expansion.
  • 34. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Roadmap for bounded expansion class Decomposition: Low tree-depth decomposition; Coloration for quantifier elimination; Building blocs: colored rooted forests with bounded height; Limit of colored rooted forests; Reconstruction of the limit building blocs; Gluing limit building blocs together.
  • 35. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Roadmap for bounded expansion class Gn G lift forget k→∞ (Gn , cn,k ) (Gk , ck ) (G, c) (Gn [I], cn,k )I∈([Nk ]) (Gk [I], ck )I∈([Nk ]) (GI , cI )I∈ k k k [Nk ] n→∞ k→∞ (Yn,k,I , cn,k , γn,k,I )I∈([Nk ]) (Yk,I , ck , γk,I )I∈([Nk ]) (YI , cI , γI )I∈ k k k [Nk ]
  • 36. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Other Voices, Other Rooms.
  • 37. Introduction Definitions Examples Theories Space Sparse Graphs Forests Bounded Expansion Other Voices, Other Rooms. Thank you for your attention.

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