Mathematical problems of very large networks Lszl Lovsz Etvs Lornd - - PowerPoint PPT Presentation

December 2008 1

Mathematical problems of very large networks

László Lovász Eötvös Loránd University, Budapest

lovasz@cs.elte.hu

December 2008 2

Issues on very large graphs The following issues are closely related:

• property testing;
• parameter estimation;
• limit objects for convergent graph sequences;
• regularity lemmas;
• distance of graphs;
• duality of left and right convergence.

December 2008 3

Issues on very large graphs The following concepts are cryptomorphic:

• a consistent local finite random graph model;
• a consistent local countable random graph;
• a measurable, symmetric function W: [0,1]2→[0,1];
• a multiplicative graph parameter with nonnegative

Möbius transform;

• a multiplicative, reflection positive graph parameter;
• A point in the completion of the set of finite graphs

with the cut-distance.

December 2008 4

Cut distance of two graphs

' 2 , ( )

| ( , ) ( , ) | ( , ') max

G G S T V G

e S T e S T d G G n

⊆

− =

• ( )

( ') V G V G =

(a) (b) |

( ) | | ( ') | V G V G n = =

* '

( , ') min ( , ')

G G

G G d G G δ

↔

=

December 2008 5

| ( ) | ' | ( ') | V G n n V G = ≠ =

(c) blow up nodes, or fractional overlay

( ( ), ( ') ) ( ')

1 ' ( 1 , )

ij i V G j V G ij ij i V G j V G

X X n n X

∈ ∈ ∈ ∈

= ≥ =

∑ ∑

, ( ) ( ' ( ) ( , ) , )

( min ma , ' ) x ') (

uv iu jv ij i u S j v X S T T V G V G

G G X X a a δ

⊆ × ∈ ∈

= = −

∑ ∑

• Cut distance of two graphs

December 2008 6

Examples:

1 , 2

1 ( , (2 , )) 8

n n

K n

• δ

≈ G

1 1 1 2 2 2

( , ), ( , ) 1)

( ) (

n n

• δ

=

G

G

1 1 1 1 2 2

( , ), ( , ), 1/ 2 1)

( ) ( ) (

n n

• δ

δ

= =

• G

G

1/2

Cut distance of two graphs

December 2008 7

Sampling Lemmas

( ) ( ') ( ) : , ': graphs with random set of nodes

k

V G V G G G k G V = ⊆ S

10 ( , [ ]) log

k

G G k δ < S

• With large probability,

Borgs-Chayes-Lovász-Sós-Vesztergombi

1/4

10 ( [ ], '[ ]) ( , ')

k k

d G G d G G k − < S S

• With large probability,

Alon-Fernandez de la Vega-Kannan-Karpinski+

December 2008 8

Regularity Lemmas Original Regularity Lemma

Szemerédi 1976

“Weak” Regularity Lemma

Frieze-Kannan 1999

“Strong” Regularity Lemma

Alon – Fischer

• Krivelevich
• M. Szegedy 2000

December 2008 9

1

{ ,..., } ( ) : ( ) ( , ) ( , ) partition of is the complete graph on with edgeweights

uv k G i j i j i j

V V V G V G e V V u V v V V G w V = = ∈ ∈ ⋅

P

P Regularity Lemmas

December 2008 10

Regularity Lemmas

1 ( ) 1 ( , ) . log For every graph and there is a partition

• f

with classes such that G k V G k G G k δ ≥ ≤

P

P

• “Weak" Regularity Lemma (Frieze-Kannan):

1 2 ( , ) . log For every graph and there is a graph with nodes such that G k H k G H k δ ≥ ≤

December 2008 11

2

: ( , ) ( ) ( ) E E E

v u su vu w tw wv

d s t a a a a = −

Fact 1. This is a metric, computable by sampling Fact 2. Weak Szemerédi partition ↔ partition most nodes into sets with small diameter “Weak" Regularity Lemma: geometric form

s t v w u

December 2008 12

“Weak" Regularity Lemma: geometric form

[0,1]: ( ) ( ) , Ex d S S d x S ⊆ = [0,1]: ( ) ( , ) partition of d G G r =

P

P P

• average ε-net

regular partition

∀S⊆[0,1] ⇒Voronoi cells of S form a partition with ∀ partition P={V1,...,Vk} of [0,1] ∃ vi∈ Vi with

( ) 8 ( ) r d S < P

1

({ ,..., }) 12 ( )

k

d v v r < P

LL – B. Szegedy

December 2008 13

• Select random nodes v1, v2, ...
• Put vi in U iff d2(vi,u)>ε for all u∈U.
• Begin with U=∅.
• Stop if for more than 1/ε2 trials, U did not grow.

Algorithm to construct representatives of classes:

size bounded by O(min # classes)

“Weak” Regularity Lemma: algorithm

December 2008 14

“Weak” Regularity Lemma: algorithm Let U={u1,...,uk}. Put a node v in Vi iff ui is the nearest node to v in U.

Algorithm to decide in which class v belongs:

December 2008 15

Max Cut in huge graphs

• Construct U as for the weak Szemerédi partition
• Compute pij = density between classes Vi and Vj

(use sampling)

• Compute max cut (U1,U2) in complete graph on U with

edge-weights pij

Algorithm to construct representation of cut:

(Different algorithm implicit by Frieze-Kannan.)

December 2008 16

Max Cut in huge graphs

• Put v∈V into V1 if d2(v,U1) ≤ d2(v,U2)

V2 if d2(v,U1) > d2(v,U2)

Algorithm to decide in which class does v belong:

December 2008 17

Convergent graph sequences

| ( )|

hom( , ) | ( ) | ( , )

V F

F G V G t F G =

Probability that random map V(F)→V(G) is a hom (i) and (ii) are equivalent.

1 2

( , ,...) ( , ) convergent: is convergent

n

G G F t F G ∀ (ii) (G1, G2,...) convergent: Cauchy in the -metric.

δ

(i) distribution of k-samples is convergent for all k

hom( , ): # of homomorphisms of into G G H H =

December 2008 18

Convergent graph sequences

with probability 1 Example: random graphs

( )

| ( )|

1 2

1 2

( , ) ( , )

E F

n t F → G ฀

1 1 2 2

( , ), ( , ) ( , )

( )

n m n m δ → → ∞ G G

December 2008 19

| ( , ) ( , ) | ( ) ( , ) t F G t F H E F G H δ − ≤

• "Counting lemma":

1 | ( , ) ( , )| t F G t F H k − ≤

“Inverse counting lemma": if

10 ( , ) log G H k δ <

• for all graphs F with k nodes, then

(i) and (ii) are equivalent.

Convergent graph sequences

December 2008 20

Limit objects

• a consistent local finite random graph model

[ ]: probability distribution on -point graphs

k

G k S

1

( \{ } a) [ ] [ ] has same distribution as

k

k

G v G

−

S S

1 2 1 2

(b) , [ ] [ ] for and areindependent. S S S G S G S = ∪

• Every random graph model with (a) and (b) is the limit of

models G[S]. consistent local

December 2008 21

• a consistent local finite random graph model
• a consistent local countable random graph

Limit objects

1 1/3 2/3 1/24 3/24 3/24 1/24

... countable random graph

December 2008 22

• a consistent local finite random graph model
• a consistent local countable random graph
• a measurable, symmetric function W: [0,1]2→[0,1]

Limit objects

December 2008 23

Limit objects

0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0

December 2008 24

A random graph with 100 nodes and with 2500 edges

1/2

Limit objects

December 2008 25

Rearranging the rows and columns

Limit objects

December 2008 26

A random graph with 100 nodes and with 2500 edges

1/2

(no matter how you reorder the nodes)

Limit objects

December 2008 27

A randomly grown uniform attachment graph with 200 nodes

1 max( , ) x y −

Limit objects

December 2008 28

( , ) : 1 max( , ) W x y x y = −

3

( , ) ( , ) ( , ) ( , )

n

t K G W x y W y z W x z dx dy dz → ∫∫∫

Limit objects

December 2008 29

{ }

2

: [0,1] [0,1] symmetric, measurable W = → W

( )

( ) [0,1]

( , ) ( , )

V F

i j ij E F

W x x dx t F W

∈

=

∏ ∫

( , ) ( , )

G

t F G t F W =

Adjacency matrix

• f graph G:

1 1 1 1 1 1 1 1 1 1 ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ ⎠

Associated function WG:

Limit objects

December 2008 30

Distance of functions

, [0,1]

( , ') inf sup ( ')

S T S T

W W W W δ

⊆ ×

= −

∫

• '

( , ') ( , )

G G

G G W W

• δ

δ =

is compact.

( , ) δ W

• Equivalent to the Regularity Lemma

Limit objects

December 2008 31

( , ) (i)

n

G

W W δ → ( ) ( , ) ( , ) (ii)

n

F t F G t F W ∀ →

Converging to a function

:

n

G W →

(i) and (ii) are equivalent.

Limit objects

December 2008 32

( , ) W x y ( , ) ( , )

n

F t F G t F W ∀ → ( , )

n

G W δ →

฀ n

G

Limit objects

December 2008 33

LL – B. Szegedy For every convergent graph sequence (Gn) there is a such that . Conversely, ∀W ∃(Gn) such that

W ∈ W

n

G W →

n

G W →

W is essentially unique (up to measure-preserving transform).

Borgs – Chayes - LL

Limit objects

December 2008 34

• a consistent local finite random graph model
• a consistent local countable random graph
• a measurable, symmetric function W: [0,1]2→[0,1]

Limit objects

2 1

:[0,1] [0,1]. ,..., [0,1] Fix Let ind uniform.

n

W X X → ∈ {1,..., } ( , ( ( , )) ( , ) ) ( )

( )

P

i j

V n W i n W X X j E n W = ∈ = G G

1/ 2 ( ,1/ 2) W n ≡ ⇒ G

W-random graphs

December 2008 35

• a consistent local finite random graph model
• a consistent local countable random graph
• a measurable, symmetric function W: [0,1]2→[0,1]
• a multiplicative graph parameter with nonnegative

Möbius transform

Limit objects

| ( ')\ ( )| ' †

( 1) ( ') ( )

E F E F F F

f f F F

⊇

= −

∑

†

( ) ( [ ]) ( ) ( [ ]) P P

k k

f F F G f F F G = ⊆ = = S S

( )

( ) [0,1]

( , ) ( , )

V F

i j ij E F

W x x dx t F W

∈

=

∏ ∫

December 2008 36

• a consistent local finite random graph model;
• a consistent local countable random graph;
• a measurable, symmetric function W: [0,1]2→[0,1];
• a multiplicative graph parameter with nonnegative

Möbius transform;

• a multiplicative, reflection positive graph parameter;

(connection matrices are positive semidefinite)

Limit objects

Many applications in extremal graph theory

December 2008 37

• a consistent local finite random graph model;
• a consistent local countable random graph;
• a measurable, symmetric function W: [0,1]2→[0,1];
• a multiplicative graph parameter with nonnegative

Möbius transform;

• a multiplicative, reflection positive graph parameter;
• A point in the completion of the set of finite graphs

with the cut-metric.

Limit objects

December 2008 38

Parameter estimation

f is estimable ⇔ f(Gn) is convergent if (Gn) is convergent

Graph parameter f is estimable:

1 ( ) . | ( [ ]) ( ) | P

k

f G f k G ε ε ε ∀ − > > ∃ ≥ < S

December 2008 39

Parameter estimation

f is estimable ⇔ (1) (2) if G(m) is obtained from G by replacing each node by m copies, then is convergent. (3)

( ) ( '), ( , ') ( ) ( ') V G V G d G G f G f G ε δ δ ε ∀ ∃ = < ⇒ − <

• (

) ( \ ) ( ) k V G k f G v f G ε ε ∀ ∃ > ⇒ − < ( ( )) f G m

Borgs, Chayes, LL, Sós, Vesztergombi