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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.