State of the art More statistics Modelings Small trees, and more A first intermediate class with limit object Jaroslav Nešetřil Patrice Ossona de Mendez Charles University CAMS, CNRS/EHESS LEA STRUCO Praha, Czech Republic Paris, France STRUCO Meeting on Distributed Computing and Graph Theory — Pont-à-Mousson — November 2013
State of the art More statistics Modelings Small trees, and more State of the art
State of the art More statistics Modelings Small trees, and more Limit objects G 4 G 2 G 3 G 1 graphon random-free graphon graphing
State of the art More statistics Modelings Small trees, and more Limit objects: dense case G 4 G 2 G 3 G 1 graphon random-free graphon graphing
State of the art More statistics Modelings Small trees, and more Graphons G W G c d b h g f e e a d c b a h f a b c d e f g h g
State of the art More statistics Modelings Small trees, and more Graphons G W G c d b h g f e e a d c b a h f a b c d e f g h g W G 1 W G 2 W G 3 . . . W G n . . . W
State of the art More statistics Modelings Small trees, and more Szemerédi partitions Regularity Lemma ∀ V ′ ∀ V ′ i ⊆ V i j ⊆ V j | V ′ i | > ǫ | V i | and | V ′ j | > ǫ | V j | ⇓ � � � dens( V ′ i , V ′ � < ǫ j ) − dens( V i , V j )
State of the art More statistics Modelings Small trees, and more Szemerédi partitions Regularity Lemma ∀ V ′ ∀ V ′ i ⊆ V i j ⊆ V j | V ′ i | > ǫ | V i | and | V ′ j | > ǫ | V j | ⇓ � � � dens( V ′ i , V ′ � < ǫ j ) − dens( V i , V j ) ǫ − → 0
State of the art More statistics Modelings Small trees, and more Szemerédi partitions Regularity Lemma ∀ V ′ ∀ V ′ i ⊆ V i j ⊆ V j | V ′ i | > ǫ | V i | and | V ′ j | > ǫ | V j | ⇓ � � � dens( V ′ i , V ′ � < ǫ j ) − dens( V i , V j )
State of the art More statistics Modelings Small trees, and more L-convergence • Convergence of δ � or of Lovász profile t ( F, G n ) = hom( F, G n ) . | G n | | F | • Limit as a graphon (Lovász–Szegedy) symmetric W : [0 , 1] × [0 , 1] → [0 , 1] (up to weak-equivalence) • Limit as an exchangeable random infinite graph (Aldous–Hoover–Kallenberg, Diaconis–Janson).
State of the art More statistics Modelings Small trees, and more Limit objects: random-free case G 4 G 2 G 3 G 1 graphon random-free graphon graphing
State of the art More statistics Modelings Small trees, and more Random-free graphons & Borel graphs Definition A graphon is random-free if it is a.e. { 0 , 1 } -valued. A Borel graph is a graph on a standard probability space, whose edge set is measurable. Connections with . . . • Vapnik–Chervonenkis dimension (Lovász-Szegedy) • δ 1 -metric (Pikhurko) • entropy (Aldous, Janson, Hatami & Norine) • class speed (Chatterjee, Varadhan)
State of the art More statistics Modelings Small trees, and more Limit objects: bounded degree case G 4 G 2 G 3 G 1 graphon random-free graphon graphing
State of the art More statistics Modelings Small trees, and more Bounded degree graphs: BS-convergence • Convergence of |{ v, B d ( G n , v ) ≃ ( F, r ) }| . | G n | • Limit as a graphing = Borel graph satisfying the Mass Transport Principle (Aldous–Lyons, Elek) � � ∀ A, B ∈ Σ d B ( x ) d ν ( x ) = d A ( x ) d ν ( x ) . A B • Limit as a unimodular distribution on rooted connected countable graphs (Benjamini–Schramm).
State of the art More statistics Modelings Small trees, and more BS-convergence µ 2 − 1 2 − 2 2 − 3 2 − 4 2 − 5 . . .
State of the art More statistics Modelings Small trees, and more Resume Dense Bounded degree L-convergence BS-convergence conv. of hom( F,G n ) conv. of |{ v, B d ( G n ,v ) ≃ ( F,r ) }| | G n | | F | | G n | Graphon Graphing Exchangeable random Unimodular distribution of infinite graph rooted connected countable graphs edge density/regularity structure
State of the art More statistics Modelings Small trees, and more More statistics
State of the art More statistics Modelings Small trees, and more Probabilistic approach of properties Definition (Stone pairing) Let φ be a first-order formula with p free variables and let G = ( V, E ) be a graph. The Stone pairing of φ and G is � φ, G � = Pr( G | = φ ( X 1 , . . . , X p )) , for independently and uniformly distributed X i ∈ V . That is: � � � { ( v 1 , . . . , v p ) ∈ V p : G | = φ ( v 1 , . . . , v p ) } � � φ, G � = , | V | p
State of the art More statistics Modelings Small trees, and more Structural Limits Definition A sequence ( G n ) is FO -convergent if, for every φ ∈ FO , the sequence � φ, G 1 � , . . . , � φ, G n � , . . . is convergent. In other words, ( G n ) is FO -convergent if, for every first-order formula φ ∈ FO , the probability that G n satisfies φ for a random assignment of the free variables converges.
State of the art More statistics Modelings Small trees, and more Structural Limits Definition Let X be a fragment of FO . A sequence ( G n ) is X -convergent if, for every φ ∈ X , the sequence � φ, G 1 � , . . . , � φ, G n � , . . . is convergent. In other words, ( G n ) is X-convergent if, for every first-order for- mula φ ∈ X, the probability that G n satisfies φ for a random assignment of the free variables converges.
State of the art More statistics Modelings Small trees, and more Special Fragments QF Quantifier free formulas L-limits FO 0 Sentences Elementary limits FO local Local formulas (BS-limits) FO All first-order formulas FO-limits
State of the art More statistics Modelings Small trees, and more Structural Limits Boolean algebra B ( X ) Stone Space S ( B ( X )) Formula φ Continuous function f φ Vertices v 1 , . . . , v p , . . . Type T of v 1 , . . . , v p , . . . Graph G statistics of types =probability measure µ G � � φ, G � f φ ( T ) d µ G ( T ) X -convergent ( G n ) weakly convergent µ G n Γ = Aut( B ( X )) Γ -invariant measure
State of the art More statistics Modelings Small trees, and more Modelings
State of the art More statistics Modelings Small trees, and more Modelings Definition A modeling A is a graph on a standard probability space s.t. every first-order definable set is measurable.
State of the art More statistics Modelings Small trees, and more Basic interpretations G = ( V, E ) �→ I ( G ) = ( V, E ′ ) E ′ = { ( x, y ) : G | = θ ( x, y ) } . Examples x �∼ y − → I ( G ) = G I ( G ) = G 2 ( x ∼ y ) ∨ ( ∃ z ( x ∼ z ) ∧ ( z ∼ y )) − →
State of the art More statistics Modelings Small trees, and more Basic interpretations G = ( V, E ) �→ I ( G ) = ( V, E ′ ) E ′ = { ( x, y ) : G | = θ ( x, y ) } . Properties ∃ I ⋆ : FO → FO , � φ, I ( G ) � = � I ⋆ ( φ ) , G �
State of the art More statistics Modelings Small trees, and more Basic interpretations G = ( V, E ) �→ I ( G ) = ( V, E ′ ) E ′ = { ( x, y ) : G | = θ ( x, y ) } . Properties ∃ I ⋆ : FO → FO , � φ, I ( G ) � = � I ⋆ ( φ ) , G � G n is FO-convergent = ⇒ I ( G n ) is FO-convergent .
State of the art More statistics Modelings Small trees, and more Basic interpretations G = ( V, E ) �→ I ( G ) = ( V, E ′ ) E ′ = { ( x, y ) : G | = θ ( x, y ) } . Properties ∃ I ⋆ : FO → FO , � φ, I ( G ) � = � I ⋆ ( φ ) , G � G n is FO-convergent = ⇒ I ( G n ) is FO-convergent . FO ⇒ I ( G n ) FO − − → A = − − → I ( A ) . G n
State of the art More statistics Modelings Small trees, and more Modelings as FO-limits? Theorem (Nešetřil, POM 2013) If a monotone class C has modeling FO -limits then the class C is nowhere dense. Nowhere dense Somewhere dense Ω( n 1+ ǫ ) edges bounded degree Ω( n 2 ) Bounded ultra sparse edges expansion minor closed
State of the art More statistics Modelings Small trees, and more Proof (sketch) • Assume C is somewhere dense. There exists p ≥ 1 such that Sub p ( K n ) ∈ C for all n ; • For an oriented graph G , define G ′ ∈ C : G x y p p p p G ′ x ′ � �� � � �� � � �� � y ′ p p � �� � � �� � (2 p + 1)( | G | − d G ( x )) − 1 (2 p + 1)( | G | − d G ( y )) − 1 p • ∃ basic interpretation I , such that for every graph G , = G [ k ( G )] def I ( G ′ ) ∼ = G + , where k ( G ) = (2 p + 1) | G | .
State of the art More statistics Modelings Small trees, and more Proof (sketch) L G n 1 / 2 G ′ n ∈ C
State of the art More statistics Modelings Small trees, and more Proof (sketch) L G n 1 / 2 FO G ′ A n
State of the art More statistics Modelings Small trees, and more Proof (sketch) L G n 1 / 2 FO G ′ A n I I FO G + I ( A ) n
State of the art More statistics Modelings Small trees, and more Proof (sketch) L G n 1 / 2 FO G ′ A n I I FO G + I ( A ) n ⇓ L G + W I ( A ) n
State of the art More statistics Modelings Small trees, and more Proof (sketch) L L G + G n 1 / 2 ⇐ ⇒ 1 / 2 n FO G ′ A n I I FO G + I ( A ) n ⇓ L G + W I ( A ) n
Recommend
More recommend
