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
Some old and new results from extremal graph theory Turán’s Theorem (special case proved by Mantel): G contains no triangles #edgesn2/4 Extremal: Theorem (Goodman): July 2010
Some old and new results from extremal graph theory Kruskal-Katona Theorem (very special case): n k July 2010
Some old and new results from extremal graph theory Semidefiniteness and extremal graph theory Tricky examples Kruskal-Katona 1 Goodman Razborov 2006 Fisher Bollobás Lovász-Simonovits 1/2 2/3 3/4 1 Mantel-Turán July 2010
Some old and new results from extremal graph theory Theorem (Erdős): G contains no 4-cycles #edgesn3/2/2 (Extremal: conjugacy graph of finite projective planes) Theorem (Chung-Graham-Wilson): Quasirandom graphs July 2010
General questions about extremal graphs Which inequalities between subgraph densities are valid? - Is there always an extremal graph? - Which graphs are extremal? July 2010
Homomorphism functions Homomorphism: adjacency-preserving map coloring independent set triangles July 2010
Homomorphism functions Probability that random map V(G)V(H) is a hom Weighted version: July 2010
Homomorphism functions Examples: hom(G, ) = # of independent sets in G if G has no loops July 2010
Homomorphism functions 3 3 -1 1/4 1/4 -1 -1 -1 -1 1/4 1/4 -1 3 3 H 1 2 H partition functions in statistical physics... July 2010
Which parameters are homomorphism functions? Graph parameter: isomorphism-invariant function on finite graphs 1 2 k-labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes 1,3 2 k-multilabeled graph: nodes labeled 1,...,k, any number of unlabeled nodes July 2010
Connection matrices ... M(f, k) k=2: ... July 2010
Which parameters are homomorphism functions? is positive semidefinite and has rank Freedman - L - Schrijver Many extensions and generalizations July 2010
k-labeled quantum graph: Computing with graphs k-labeled quantum graph: finite formal sum of k-labeled graphs 1 2 infinite dimensional linear space July 2010
is a commutative algebra with unit element Computing with graphs Define products: is a commutative algebra with unit element ... July 2010
f: graph parameter Computing with graphs Inner product: extend linearly July 2010
Computing with graphs Factor out the kernel: July 2010
Computing with graphs Example 1: - - f( ) f( ) = 0 July 2010
f( ) f( ) f( ) = 0 Computing with graphs if is an integer Example 2: - (-1) + - (-1) + f( ) f( ) f( ) = 0 July 2010
f is reflection positive Computing with graphs f is reflection positive July 2010
Computing with graphs Write if for every graph H . Turán: -2 + Kruskal-Katona: - Blakley-Roy: - Sidorenko Conjecture: (F bipartite) July 2010
- Computing with graphs -4 +2 - + 2 = - + 2 = 2 - + = + + ≥ 0 - 2 + - 2 ≥ 0 t( ,G) – 2t( ,G) + t( ,G) ≥ 0 Goodman’s Theorem July 2010
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! Hatami-Norine is algorithmically undecidable. July 2010
The main trick in the proof Kruskal-Katona 1 Goodman Razborov 2006 Fisher Bollobás Lovász-Simonovits 1/2 2/3 3/4 1 Mantel-Turán July 2010
A weak Positivestellensatz July 2010
Which inequalities between densities are valid? Undecidable, but decidable with an arbitrarily small error. July 2010
Is there always an extremal graph? Minimize over x0 Real numbers are useful minimum is not attained in rationals Minimize t(C4,G) over graphs with edge-density 1/2 always >1/16, arbitrarily close for random graphs Quasirandom graphs Graph limits are useful minimum is not attained among graphs July 2010
Pixel pictures AG G WG July 2010 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 AG G WG July 2010
Limit objects July 2010
Limit objects 1/2 A random graph with 100 nodes and with 2500 edges July 2010
Limit objects Rearranging the rows and columns July 2010
Limit objects A randomly grown uniform attachment graph with 200 nodes July 2010
Limit objects (graphons) July 2010
W is essentially unique (up to 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). BCL July 2010
Semidefinite connection matrices is positive semidefinite, f( )=1 and f is multiplicative is positive semidefinite July 2010
Proof of the weak Positivstellensatz (sketch2) The optimum of the semidefinite program minimize subject to M(f,k) positive semidefinite for all k f(K1)=1 is 0. Apply the Duality Theorem of semidefinite programming July 2010
W is essentially unique (up to 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). BCL July 2010
Is there always an extremal graph? No, but there is always an extremal graphon. July 2010