1. Around the Regularity Lemma
László Lovász, Balázs szegedy
Microsoft Research IAS Princeton

2. 1. Szemerédi’s Regularity Lemma
2. The Regularity Lemma in Hilbert space
3. The Regularity Lemma as compactness
4. The Regularity Lemma as dimensionality

3. 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).
Szemerédi partition with
error 
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

4. X Y

5. Regularity Lemma Light 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

6. Strong Regularity Lemma
Alon, Fischer, Krivelevich, Szegedy
(0,1,...,k,...)  k0>0 such that G we can
change at most 0|V(G)|2 edges so that the
resulting graph G' has an equipartition Q=(V1,... ,Vk) (kk0)
s.t. 1i j k, XVi, YVj,

7. A lemma about Hilbert space
B.Szegedy
Corollary: approximation by stepfunction

8. Stepfunction approximation  Weak regularity lemma
Adjacency matrix of G, viewed as a function
Strong lemma
Weak lemma
Stepfunction approximation  Weak regularity lemma

9. Rectangle norm:
Rectangle distance:

10. Weak Regularity Lemma:
is compact
L-Szegedy

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

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

13. Approximate uniqueness
Borgs-Chayes-
L-T.Sós-Vesztergombi
If G1 and G2 are graphs on n nodes so that for all F with
then G1 and G2 can be overlayed so that for all

14. Applications:
- Limits of graph sequences
- Graph parameter testing
- Extremal graph theory

15. A random graph
with 100 nodes and with 2500 edges

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

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

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

19. For a sequence of graphs (Gn), the following are equivalent:
(iii)
(iii)
random graphs
uniform attachment graphs
preferential attachment graphs

20. -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)?

21. 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.

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

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

24. ...
k=2:
...

25. f is moment parameter L-Szegedy
Gives inequalities between subgraph densities

26. Proof of Kruskal-Katona