Graph algebras and graph limits Lszl Lovsz Etvs

Graph algebras and graph limits László Lovász Eötvös Loránd University, Budapest Joint work with Christian Borgs, Jennifer Chayes, Balázs Szegedy, Vera Sós and Katalin Vesztergombi July 2010 1

Some old and new results from extremal graph theory Turán’s Theorem (special case proved by Mantel): G contains no triangles #edges n 2/4 Extremal: Theorem (Goodman): July 2010 2

Some old and new results from extremal graph theory Kruskal-Katona Theorem (very special case): k July 2010 n 3

Semidefiniteness and new extremal graph theory Some old and results from Trickytheory examples extremal graph Kruskal-Katona 1 Razborov 2006 Fisher Goodman 0 1/2 2/3 3/4 Bollobás Mantel-Turán July 2010 1 Lovász-Simonovits 4

Some old and new results from extremal graph theory Theorem (Erdős): G contains no 4 -cycles #edges n 3/2/2 (Extremal: conjugacy graph of finite projective planes) Theorem (Chung-Graham-Wilson): m o d gra s h p n a r si a u Q July 2010 5

General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Is there always an extremal graph? - Which graphs are extremal? July 2010 6

Homomorphism functions Homomorphism: adjacency-preserving map coloring independent set triangles July 2010 7

Homomorphism functions Probability that random map V(G) V(H) is a hom Weighted version: July 2010 8

Homomorphism functions Examples: hom(G, ) = # of independent sets in G if G has no loops July 2010 9

Homomorphism functions 3 3 -1 1/4 -1 -1 -1 1/4 -1 3 H 3 1 1 H 2 1 1 partition functions in statistical physics. . . July 2010 10

Which parameters are homomorphism functions? Graph parameter: isomorphism-invariant function on finite graphs k-labeled graph: k nodes labeled 1, . . . , k, any number of unlabeled nodes k-multilabeled graph: nodes labeled 1, . . . , k, any number of unlabeled nodes July 2010 1 2 1, 3 2 11

Connection matrices M(f, k) k=2: . . . July 2010 12

Which parameters are homomorphism functions? is positive semidefinite and has rank Freedman - L - Schrijver Many extensions and generalizations July 2010 13

Computing with graphs k-labeled quantum graph: 1 finite formal sum of k-labeled graphs 2 infinite dimensional linear space July 2010 14

Computing with graphs Define products: is a commutative algebra with unit element July 2010 . . . 15

Computing with graphs Inner product: f: graph parameter extend linearly July 2010 16

Computing with graphs Factor out the kernel: July 2010 17

Computing with graphs Example 1: - f( July 2010 ) - f( )=0 18

Computing with graphs if is an integer Example 2: - ( -1) f( July 2010 + ) - ( -1) f( ) + f( )=0 19

Computing with graphs f is reflection positive July 2010 20

Computing with graphs Write if Turán: -2 for every graph H. + - Kruskal-Katona: Blakley-Roy: - Sidorenko Conjecture: July 2010 (F bipartite) 21

Computing with graphs 2 - - + = 2 - - 2 +2 + - 2 = -2 t( + , G) – 2 t( July 2010 ≥ 0 , G) + t( , G) ≥ 0 - - =2 -2 +2 -4 - + + +2 -4 +2 + Goodman’s Theorem 22

Positivstellensatz for graphs? If a quantum graph x is sum of squares (modulo labels and isolated nodes), then Question: Suppose that Does it follow that . No! is algorithmically undecidable. Hatami-Norine July 2010 23

The main trick in the proof Kruskal-Katona 1 Razborov 2006 Fisher Goodman 0 1/2 2/3 3/4 Bollobás Mantel-Turán July 2010 1 Lovász-Simonovits 24

A weak Positivestellensatz July 2010 25

Which inequalities between densities are valid? Undecidable, but decidable with an arbitrarily small error. July 2010 26

Is there always an extremal graph? Minimize over x 0 minimum is not attained in rationals Real numbers are useful Minimize t(C 4, G) over graphs with edge-density 1/2 July 2010 s h p a r g alwaysm>1/16, o d n arbitrarily close a for random graphs r i as u Q Graph limits minimum is not attained are useful among graphs 27

Pixel pictures G 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 AG WG July 2010 28

Limit objects July 2010 29

Limit objects A random graph with 100 nodes and with 2500 edges July 2010 1/2 30

Limit objects Rearranging the rows and columns July 2010 31

Limit objects A randomly grown uniform attachment graph with 200 nodes July 2010 32

Limit objects (graphons) July 2010 33

Limit objects For every convergent graph sequence (Gn) there is a graphon such that LS Conversely, for every graphon W there is a graph sequence (Gn) such that LS W is essentially unique (up to measure-preserving transformation). July 2010 BCL 34

Semidefinite connection matrices is positive semidefinite, f( )=1 and f is multiplicative is positive semidefinite July 2010 35

Proof of the weak Positivstellensatz (sketch 2) The optimum of the semidefinite program minimize subject to M(f, k) positive semidefinite for all k is 0. f(K 1)=1 Apply the Duality Theorem of semidefinite programming July 2010 36

Limit objects For every convergent graph sequence (Gn) there is a graphon such that LS Conversely, for every graphon W there is a graph sequence (Gn) such that LS W is essentially unique (up to measure-preserving transformation). July 2010 BCL 37

Is there always an extremal graph? No, but there is always an extremal graphon. July 2010 38