1. Graph limits and graph homomorphisms László Lovász Microsoft Research

2. 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?

3. 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

4. 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:

5. Limits of graph sequences
Which sequences are convergent?
Is there a limit object?
Which parameters are “continuous at infinity”?

6. Homomorphism: adjacency-preserving map
coloring
independent
set
triangles

7. Probability that random map
V(G)V(H) is a hom
Weighted version:

8. Examples:
hom(G, ) = # of independent sets in G

9. Which graph sequences are convergent?
Example: random graphs
with probability 1

10. (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

11. (Gn) is convergent  Cauchy in the -metric.
Distance of graphs
(Gn) is convergent  Cauchy in the metric.
"Counting lemma":

12. Approximating by small graphs
Szemerédi's Regularity Lemma
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

13. X Y

14. 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

15. Corollary of the "weak" Regularity Lemma:

16. Limits of graph sequences
Which sequences are convergent?
(G1, G2,...) convergent  Cauchy in the metric.
Is there a limit object?

17. 1/2
A random graph
with 100 nodes and with 2500 edges

18. A randomly grown
uniform attachment graph
with 100 nodes born at random times
and with 2500 edges

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

20. A randomly grown
preferential attachment graph
with 100 fixed nodes (ordered by degrees)
and with 5,000 edges

21. The limit object as a function

22. 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

23. Distance of functions

24. Restatement of the "Weak" Regularity Lemma:

25. 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).

26. The limit object as a graph parameter
is a graph parameter
(normalized)
(multiplicative)
"connection matrices" are positive semidefinite
(reflection positive)

27. ...
k=2:

28. Gives inequalities between subgraph densities
 extremal graph theory

29. The limit object as a random graph model
W-random graphs:

30. 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

31. -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)?

32. [(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.

33. max cut