1. Partitioning and decomposing graphs László Lovász
Hungarian Academy of Sciences
and
Eötvös Loránd University, Budapest
January 2018

2. Graph partition problems
„Aggragate and
scale down”
Regularity partitions
Representative sets
Sampling
„Divide and conquer”
Components
Blocks
Tree-decomposition
January 2018

3. Representative set of nodes
Small size, (almost) every node is “similar” to
one of the nodes in the set
When are two nodes similar?
- Neighbors...
- (Almost) same neighborhood...
- (Almost) same second neighborhood…
For dense graphs,
the „right” notion 3

4. Representative set of nodes
random edge set, with density 1/2 4

5. 2-neighborhood representation
A=AG: adjacency matrix of G
n=|V(G)|: number of nodes
Columns of A2: representation in Rn
Columns of A: representation in Rn
October 2015

6. 2-neighborhood representation
A: adjacency matrix of graph G
ai: column i of A2
12/30/2018 6

7. 2-neighborhood representation
number of 2-paths:
highly concentrated
around expectation
probability of
connecting
f(distance)
2-neighborhood distance: ~ spherical distance
2-neighborhood representation: ~ sphere
October 2015

8. Weak Regularity Lemma (Szemerédi) Frieze-Kannan
Regularity partitions
Weak Regularity Lemma (Szemerédi) Frieze-Kannan Vi
pij : edge density
betweenVi and Vj Vj
replace edges between Vi and Vj
by random edges with prob pij
to get GP 8

9. Weak Regularity Lemma (Szemerédi) Frieze-Kannan
Regularity partitions
Weak Regularity Lemma (Szemerédi) Frieze-Kannan S Vi
Is this a strong enough measure of closeness? Vj 9

10. Weak Regularity Lemma (Szemerédi) Frieze-Kannan
Regularity partitions
Weak Regularity Lemma (Szemerédi) Frieze-Kannan Vi Vj
# of triangles, pentagons, … (almost) same in G and GP
maximum cut, bisection, … (almost) same in G and GP
energies, entropies, … (almost) same in G and GP 10

11. 2-neighborhood representation
Voronoi diagram
= (weak) regularity partition 11

12. Using a representative setw u
This is a metric, computable by sampling.
Service
- query from node: which class am I in?
- reply: index of closest representative node 12

13. A graph partition problem
k-edge-separator: TE(G), component of G-T
has  k nodes
January 2018

14. A graph partition problem
Polynomially solvable
(but difficult)
Edmonds Matching
Algorithm
k=2: matching problem
k=3: triangle matching problem
NP-hard
Approximation?
January 2018

15. Very large graphs with bounded degree
Find a perfect matching in G.
distributed algorithm: Nguyen and Onak
January 2018

16. Very large graphs with bounded degree
distributed algorithms
local algorithms
January 2018

17. Very large graphs with bounded degree
distributed algorithms
local algorithms
property testing
January 2018

18. Fractional graph partition problem
T: optimal k-edge-separator
January 2018

19. Fractional graph partition problem
probability distribution
„marginal” uniform
expected expansion
January 2018

20. Fractional graph partition problem
Define
 probability distribution on Rk
with uniform marginal
no dependence on k or n
January 2018

21. Greedy algorithm: For j=1,2,..., select Y1,Y2,...k
Proof sketch
Greedy algorithm: For j=1,2,..., select Y1,Y2,...k
so thatYj is the minimizer of
Output: X=Y1 Y2 ...
for small k and large graph:
very many steps!
Phases, whose number is
O(1/sepk*)
January 2018

22. Very large graphs with bounded degree
distributed algorithms
local algorithms
property testing
The infinite is a good approximation
of the large finite.
January 2018

23. Bounded degree ( D) Borel graph on [0,1]
Graphing
Bounded degree ( D) Borel graph on [0,1]
with “double counting” condition:
Extends to measure  on Borel subsets of V2.
„edge measure”
January 2018

24. Why graphings? Ergodic theory Finitely generated groups
Limits of convergent graphs sequences
with bounded degree
January 2018

25. Graphing, examples
January 2018

26. Graphing, examples unit circumference  irrational components:
2-way infinite paths
January 2018

27. , irrational, lin. indep 1x1 torus , irrational
Graphing, examples
unit circumference
, irrational, lin. indep  
1x1 torus
, irrational
components: grids  
components: grids
January 2018

28. Penrose tilings
rhombic icosahedron
de Bruijn - Bárász
January 2018

29. distributed algorithms
Why graphings?
distributed algorithms
local algorithms (on bounded degree graphs)
property testing (on bounded degree graphs)
local algorithms on graphings
January 2018

30. Graphing partition problem
k-edge-separator: TE(G), component of G-T
has  k nodes
Graphing G hyperfinite: sepk(G)0 (k)
January 2018

31. Hyperfinite graphings, examples 
Diophantine approximation  
January 2018

32. Hyperfinite graph families
Family G of finite graphs is hyperfinite:
Hyperfinite: paths, trees, planar graphs,
every non-trivial minor-closed property
Non-hyperfinite: expanders
January 2018

33. {Gn} is hyperfinite  G is hyperfinite
Hyperfinite graph sequences
If Gn  G, then
{Gn} is hyperfinite  G is hyperfinite
Schramm
(Benjamini-Shapira-Schramm)
January 2018

34. Hyperfinite graph families
Every graph property is testable in any family
of hyperfinite graphs.
Many graph properties are
polynomial time testable for
graphs with bounded tree-width.
Newman – Sohler
(Benjamini-Schramm-Shapira,
Elek)
January 2018

35. Thanks, that’s all!
January 2018

36. Local equivalence, definition
xV  Gx: component of x
uniform random x  Gx: random connected
rooted (countable)
graph
unimodular random
network
Benjamini - Schramm
January 2018

37. Hyperfinite graphings
If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G
January 2018

38. Local equivalence, definition
xV  Gx: component of x
G1, G2 locally equivalent:
for random uniform xV,
distributions of (G1)x and (G2)x
are the same
January 2018

39. Local isomorphism, definition
 : V(G1)  V(G2) local isomorphism:
measure preserving and
(x) isomorphism between
(G1)x and (G2)(x)
Existence of local isomorphism
proves local equivalence.
January 2018

40. Local isomorphism, example 
(x,y)  x+y mod 1   
components: grids
components: grids
January 2018

41. G1 and G2 are locally equivalent 
Local equivalence
G1 and G2 are locally equivalent 
G and local isomorphisms GG1, GG2. G1 G2 G
January 2018

42. ? Hyperfinite graphings If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G ?
January 2018

43. Local isomorphism forward
Diophantine approximation 
(x,y)  x+y mod 1    
January 2018

44. Pushing forward and pulling back
measure
preserving
subset
linear
relaxation
measure
January 2018

45. Fractional separation duality
If G1 and G2 are locally equivalent, then G1 G2
easy ?
Duality!
January 2018