Random graphs and limits of graph sequences László Lovász Microsoft Research

W -random graphs

Adjacency matrix of weighted graph G, viewed as a function in W 0 : W G -random graphs  generalized random graphs with model G

density of F in W

Convergent graph sequences (G n ) is convergent: Examples: Paley graphs (quasirandom) half-graphs closest neighbor graphs...

Does a convergent graph sequence have a limit? For every convergent (G n ) there is a function W  W 0 such that B.Szegedy-L GnWGnW a.s.

Uniqueness of the limit Borgs-Chayes-L WWW W WW WWW W W W W W

A random graph with 100 nodes and 2500 edges 1/2 Quasirandom  converges to 1/2

Growing uniform attachment graph If there are n nodes - with prob c/n, a new node is added, - with prob (n-c)/n, a new edge is added.

A growing uniform attachment graph with 200 nodes and edges

Fixed preferential attachment graph Fix n nodes For m steps choose 2 random nodes independently with prob proportional to (deg+1) and connect them

A preferential attachment graph with 100 fixed nodes and with 5,000 (multiple) edges

A preferential attachment graph with 100 fixed nodes ordered by degrees and with 5,000 edges

Moments 1-variable functions 2-variable functions These are independent quantities. These are independent quantities. Erdős- L- Spencer Moments determine the function up to measure preserving transformation. Moment sequences are characterized by semidefiniteness Moments determine the function up to measure preserving transformation. Borgs- Chayes- L Moment graph parameters are characterized by semidefiniteness L- Szegedy Except for multiplicativity over disjoint union:

k -labeled graph: k nodes labeled 1,...,k Connection matrix of graph parameter f Connection matrices

k=2:...

f is a moment parameter L-Szegedy Gives inequalities between subgraph densities  extremal graph theory f is reflection positive

Kruskal-Katona Theorem for triangles: Turán’s Theorem for triangles: Graham-Chung-Wilson Theorem about quasirandom graphs: Extremal graph theory as properties of

k=2 Proof of Kruskal-Katona

Moments 1-variable functions 2-variable functions These are independent quantities. These are independent quantities. Erdős- L- Spencer Moments determine the function up to measure preserving transformation. Moment sequences are characterized by semidefiniteness Moments determine the function up to measure preserving transformation. Borgs- Chayes- L Moment graph parameters are characterized by semidefiniteness L- Szegedy Moment sequences are interesting Moment graph parameters are interesting

partition functions, homomorphism functions,... L-Szegedy

The following are cryptomorphic: functions in W 0 modulo measure preserving transformations reflection positive and multiplicative graph parameters f with f(K 1 )=1 random graph models G ( n ) that are - label-independent - hereditary - independent on disjoint subsets countable random graphs G that are - label-independent - independent on disjoint subsets

Rectangle norm: Rectangle distance: The structure of W 0

Weak Regularity Lemma: is compact L-Szegedy Frieze-Kannan

For a sequence of graphs (G n ), the following are equivalent: (i) (iii) uniform attachment graphs preferential attachment graphs random graphs

Approximate uniqueness Borgs-Chayes- L-T.Sós-Vesztergombi If G 1 and G 2 are graphs on n nodes so that for all F with then G 1 and G 2 can be overlayed so that for all

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)?

For a graph parameter f, the following are equivalent: (i) f can be computed by local tests (ii) (iii) f is unifomly continuous w.r.t Density of maximum cut is testable. Borgs-Chayes- L-T.Sós-Vesztergombi

Key fact: