Graph limits and graph homomorphisms László Lovász Microsoft Research lovasz@microsoft.com
Why define limits of graph sequences? I. Very large graphs: Internet -Social networks Ecological systems VLSI Statistical physics Brain Is there a good "small" approximation? Is there a good "continuous" approximation?
arbitrarily close for random graphs II. Real numbers Minimize minimum is not attained in rationals Minimize density of 4-cycles in a graph with edge-density 1/2 always >1/16, arbitrarily close for random graphs minimum is not attained among graphs
Limits of sequences of graphs with bounded degree: Aldous, Benjamini-Schramm, Lyons, Elek Borgs, Chayes, L, Sós, B.Szegedy, Vesztergombi Limits of sequences of dense graphs:
Limits of graph sequences Which sequences are convergent? Is there a limit object? Which parameters are “continuous at infinity”?
Homomorphism: adjacency-preserving map coloring independent set triangles
Probability that random map V(G)V(H) is a hom Weighted version:
Examples: hom(G, ) = # of independent sets in G
Which graph sequences are convergent? Example: random graphs with probability 1
(Gn: n=1,2,...) is quasirandom: Quasirandom graphs Thomason Chung – Graham – Wilson (Gn: n=1,2,...) is quasirandom: Example: Paley graphs p: prime 1 mod 4
(Gn) is convergent Cauchy in the -metric. Distance of graphs (Gn) is convergent Cauchy in the -metric. "Counting lemma":
Approximating by small graphs Szemerédi's Regularity Lemma 1974 Given >0 The nodes of graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities). difference at most 1 with k2 exceptions for subsets X,Y of parts Vi,Vj # of edges between X and Y is pij|X||Y| (n/k)2
X Y
Weak Regularity Lemma Frieze-Kannan 1989 Given >0 The nodes of graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities). difference at most 1 for subset X of V, # of edges in X is
Corollary of the "weak" Regularity Lemma:
Limits of graph sequences Which sequences are convergent? (G1, G2,...) convergent Cauchy in the -metric. Is there a limit object?
1/2 A random graph with 100 nodes and with 2500 edges
A randomly grown uniform attachment graph with 100 nodes born at random times and with 2500 edges
A randomly grown preferential attachment graph with 100 fixed nodes and with 5,000 (multiple) edges
A randomly grown preferential attachment graph with 100 fixed nodes (ordered by degrees) and with 5,000 edges
The limit object as a function
t(F,W)= 2-|E(F)| # of eulerian orientations of F Example 1: Associated function WG: 1 Adjacency matrix of graph G: Example 2: t(F,W)= 2-|E(F)| # of eulerian orientations of F
Distance of functions
Restatement of the "Weak" Regularity Lemma:
Summary of main results For every convergent graph sequence (Gn) there is a such that Szemerédi Lemma Conversely, W (Gn) such that W is essentially unique (up to measure-preserving transform).
The limit object as a graph parameter is a graph parameter (normalized) (multiplicative) "connection matrices" are positive semidefinite (reflection positive)
... k=2:
Gives inequalities between subgraph densities extremal graph theory
The limit object as a random graph model W-random graphs:
The following are cryptomorphic: functions in W0 modulo measure preserving transformations normalized, multiplicative and reflection positive graph parameters random graph models G(n) that are hereditary and independent on disjoint subsets ergodic invariant measures on
-Does it have an even number of nodes? Local testing for global properties What to ask? -Does it have an even number of nodes? -Is it connected? -How dense is it (average degree)?
[(Gn) convergent f(Gn) convergent] f is testable: Sk: random set of k nodes f is testable [(Gn) convergent f(Gn) convergent] Goldreich - Goldwasser - Ron The density of the largest cut can be estimated by local tests.
max cut