Graph convergence Graphons and finitely forcible graphons Universal Construction Finitely forcible graph limits are universal Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins University of Warwick Monash University - Discrete Maths Research Group Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Graph limits Approximate asymptotic properties of large graphs Extremal combinatorics/computer science : flag algebra method, property testing large networks, e.g. the internet, social networks... The ‘limit’ of a convergent sequence of graphs is represented by an analytic object called a graphon Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H complete graphs K n Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H complete graphs K n Erd˝ os-R´ enyi random graphs G n , p Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H complete graphs K n Erd˝ os-R´ enyi random graphs G n , p any sequence of sparse graphs Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H complete graphs K n Erd˝ os-R´ enyi random graphs G n , p any sequence of sparse graphs Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Dense graph convergence Convergence for dense graphs ( | E | = Ω ( | V | 2 )) d ( H , G ) = probability | H | -vertex subgraph of G is H A sequence ( G n ) n 2 N of graphs is convergent if d ( H , G n ) converges for every H complete graphs K n Erd˝ os-R´ enyi random graphs G n , p any sequence of sparse graphs Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Limit object: graphon Graphon: measurable function W : [0 , 1] 2 → [0 , 1], s.t. W ( x , y ) = W ( y , x ) ∀ x , y ∈ [0 , 1] W -random graph of order n : n random points x i ∈ [0 , 1], edge probability W ( x i , x j ) Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Limit object: graphon Graphon: measurable function W : [0 , 1] 2 → [0 , 1], s.t. W ( x , y ) = W ( y , x ) ∀ x , y ∈ [0 , 1] W -random graph of order n : n random points x i ∈ [0 , 1], edge probability W ( x i , x j ) d ( H , W ) = probability W -random graph of order | H | is H Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Limit object: graphon Graphon: measurable function W : [0 , 1] 2 → [0 , 1], s.t. W ( x , y ) = W ( y , x ) ∀ x , y ∈ [0 , 1] W -random graph of order n : n random points x i ∈ [0 , 1], edge probability W ( x i , x j ) d ( H , W ) = probability W -random graph of order | H | is H W is a limit of ( G n ) n 2 N if d ( H , W ) = lim n !1 d ( H , G n ) ∀ H Every convergent sequence of graphs has a limit W -random graphs converge to W with probability one Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Examples of graph limits The sequence of complete bipartite graphs, ( K n , n ) n 2 N 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 The sequence of random graphs, ( G n , 1 / 2 ) n 2 N 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Finitely forcible graphons A graphon W is finitely forcible if ∃ H 1 . . . H k s.t d ( H i , W 0 ) = d ( H i , W ) = ⇒ d ( H , W 0 ) = d ( H , W ) ∀ H 1. Thomason (87), Chung, Graham and Wilson (89) 2. Lov´ asz and S´ os (2008) 3. Diaconis, Holmes and Janson (2009) 4. Lov´ asz and Szegedy (2011) Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Motivation Conjecture (Lov´ asz and Szegedy, 2011) The space of typical vertices of a finitely forcible graphon is compact. Conjecture (Lov´ asz and Szegedy, 2011) The space of typical vertices of a finitely forcible graphon is finite dimensional. Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Motivation Conjecture (Lov´ asz and Szegedy, 2011) The space of typical vertices of a finitely forcible graphon is compact. Theorem (Glebov, Kr´ al’, Volec, 2013) T ( W ) can fail to be locally compact Conjecture (Lov´ asz and Szegedy, 2011) The space of typical vertices of a finitely forcible graphon is finite dimensional. Theorem (Glebov, Klimoˇ sov´ a, Kr´ al’, 2014) T ( W ) can have a part homeomorphic to [0 , 1] 1 Theorem (Cooper, Kaiser, Kr´ al’, Noel, 2015) ∃ finitely forcible W such that every ε -regular partition has at least 2 ε − 2 / log log ε − 1 parts (for inf. many ε → 0). Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Previous Constructions A A 0 B B 0 B 00 C C 0 D Q A B C D E F G P R A A B A 0 C D B E B 0 F B 00 G P C Q C 0 D R Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Universal Construction Theorem Theorem (Cooper, Kr´ al’, M.) Every graphon is a subgraphon of a finitely forcible graphon. Existence of a finitely forcible graphon that is non-compact, infinite dimensional, etc For every non-decreasing function f : R → R tending to ∞ , ∃ finitely forcible W and positive reals ε i tending to 0 such that every weak ε i -regular partition of W has at least ε − 2 ✓ ◆ i Ω f ( ε − 1 2 ) parts. i Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
Graph convergence Graphons and finitely forcible graphons Universal Construction Ingredients of the proof Partitioned graphons vertices with only finitely many degrees parts with vertices of the same degree Decorated constraints method for constraining partitioned graphons density constraints rooted in the parts based on notions related to flag algebras Encoding a graphon as a real number in [0 , 1] forcing W by fixing its density in dyadic subsquares Jacob Cooper Dan Kr´ al’ Ta´ ısa Martins Finitely forcible graph limits are universal
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