Graph limits Graph convergence Approximate asymptotic properties of large graphs Extremal combinatorics/computer science : flag algebra method, proper

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1 Jacob Cooper Dan Král Taísa Martins University of Warwick Monash University - Discrete Maths Research Group

2 Graph limits Graph convergence 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

3 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H

4 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H complete graphs K n

5 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H complete graphs K n Erdős-Rényi random graphs G n,p

6 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H complete graphs K n Erdős-Rényi random graphs G n,p any sequence of sparse graphs

7 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H complete graphs K n Erdős-Rényi random graphs G n,p any sequence of sparse graphs

8 Dense graph convergence Convergence for dense graphs ( E = ( V 2 )) d(h, G) = probability H -vertex subgraph of G is H Asequence(G n ) n2n of graphs is convergent if d(h, G n ) converges for every H complete graphs K n Erdős-Rényi random graphs G n,p any sequence of sparse graphs

9 Limit object: graphon Graphon: measurable function W :[, ] 2! [, ], s.t. W (x, y) =W (y, x) 8x, y 2 [, ] W -random graph of order n: n random points x i 2 [, ], edge probability W (x i, x j )

10 Limit object: graphon Graphon: measurable function W :[, ] 2! [, ], s.t. W (x, y) =W (y, x) 8x, y 2 [, ] W -random graph of order n: n random points x i 2 [, ], edge probability W (x i, x j ) d(h, W ) = probability W -random graph of order H is H

11 Limit object: graphon Graphon: measurable function W :[, ] 2! [, ], s.t. W (x, y) =W (y, x) 8x, y 2 [, ] W -random graph of order n: n random points x i 2 [, ], 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 ) n2n if d(h, W )= lim n! d(h, G n) 8 H Every convergent sequence of graphs has a limit W -random graphs converge to W with probability one

12 Examples of graph limits The sequence of complete bipartite graphs, (K n,n ) n2n The sequence of random graphs, (G n,/2 ) n2n

13 Finitely forcible graphons A graphon W is finitely forcible if 9 H...H k s.t d(h i, W )=d(h i, W ) =) d(h, W )=d(h, W ) 8 H. Thomason (87), Chung, Graham and Wilson (89) 2. Lovász and Sós (28) 3. Diaconis, Holmes and Janson (29) 4. Lovász and Szegedy (2)

14 Motivation Graph convergence Conjecture (Lovász and Szegedy, 2) The space of typical vertices of a finitely forcible graphon is compact. Conjecture (Lovász and Szegedy, 2) The space of typical vertices of a finitely forcible graphon is finite dimensional.

15 Motivation Graph convergence Conjecture (Lovász and Szegedy, 2) The space of typical vertices of a finitely forcible graphon is compact. Theorem (Glebov, Král, Volec, 23) T (W ) can fail to be locally compact Conjecture (Lovász and Szegedy, 2) The space of typical vertices of a finitely forcible graphon is finite dimensional. Theorem (Glebov, Klimošová, Král, 24) T (W ) can have a part homeomorphic to [, ] Theorem (Cooper, Kaiser, Král, Noel, 25) 9 finitely forcible W such that every "-regular partition has at least 2 " 2 / log log " parts (for inf. many "! ).

16 Previous Constructions A A B B B C C D A B C D E F G P Q R A A B A C B B D E F B G P C Q C D R

17 Theorem Theorem (Cooper, Král, 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, 9 finitely forcible W and positive reals " i tending to such that every weak " i -regular partition of W has at least 2 " 2 i f (" i ) parts.

18 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 [, ] forcing W by fixing its density in dyadic subsquares

19 Agraphonasarealnumber Unique representation by densities on dyadic squares 4-tuple map :(d, s, t, k)!{, } dyadic square: s, s+ 2 d 2 t, t+ d 2 d 2 d k-th bit in the standard binary representation of the density of W in the dyadic square, otherwise ' : N 4! N (bijection), : W![, ] (W ) j-th bit = (' (j))

20 Sketch of the construction A A B C D E F G P Q R B C D reference to 4-tuples (W F ) bijection ' E F G reference to dyadic squares WF P Q auxiliary structures R

21 Universal construction A B C D E F G P Q R A B C D E F G WF P Q R

22 Universality Meager set Theorem (Cooper, Král, M.) Every graphon is a subgraphon of a finitely forcible graphon. Theorem (Lovász and Szegedy, 2) Finitely forcible graphons form a meager set in the space of all graphons. Analogy: : W![, ] N (injection) S [, ] measurable (W [S S]): projection of (W )inasubspace of [, ] N e.g. H = {(C(x, y), z) (x, y, z) 2 R 3, C is a space-filling curve}

23 Thank you for your attention!