Limits of randomly growngraph sequences

Katalin VesztergombiEötvös University, Budapest

With: Christian Borgs, Jennifer Chayes, László Lovász, Vera Sós

Convergent graph sequences

1 2( , ,...) ( , )convergent: isconvergentnG G F t F G"

with probability 1

Example: random graphs

( )| ( )|12

12( , )( , )

E Fnt F ®Gt

| ( )|

hom( , )| ( ) |

( , ) V F

F GV G

t F G Probability that random mapV(F)V(G) is a hom

The limit object

{ }20 : [0,1] [0,1] symmetric, measurableW= ®W

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F WÎ

= Õò( , ( ): ) ,nn F t F G WW tG F® " ®

For every convergent graph sequence (Gn)there is a such that0W Î W nG W®

Lovász-Szegedy

0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0

Half-graphs

A random graph with

100 nodes and with edge density 1/2W1/2

Rearranging the rows and columns

A random graph with

100 nodes and with edge density 1/2W1/2

(no matter how you reorder the nodes)

Randomly grown uniform attachment graph

At step n:- a new node is born;- any two nodes are connected with probability 1/n

Ignore multiplicity of edges

A randomly grown

uniform attachment graph

with 200 nodes

probability that nodes i < j are not connected:

1 1 11

j j n jj j n n- - -=+ L

expected degree of j:

1 ( 1)( 2)2 2

n j jn

- - --

expected number of edges:2 16

n -

After n steps:

probability that nodes i and j are connected:

max( , ) 11 i jn

--

The limit:

if i=xn and j=yn

1 max( , )x y» -

These are independent events for different i,j.

A randomly grown

uniform attachment graph

with 200 nodes

( , ) 1 max( , )W x y x y= -

Proof: By estimating the cut-distance.

Randomly grown prefix attachment graph

At step n:- a new node is born;- connects to a random previous node and all its predecessors

A randomly grown prefix attachment graph

with 200 nodes

Is this graph sequenceconvergent at all?

Yes, by computing subgraph densities!

This tends to some shades of gray; is that the limit?

No, by computing triangle densities!

A randomly grown prefix attachment graph

with 200 nodes (ordered by degrees)

This also tends to some shades of gray; is that the limit?

No…

Label node born in step k, connecting to {1,…,m}, by (k/n, m/k)

1 22

1 22

21 2 ork k m k m

kx y

n nx

æ ö÷ç ÷£ £ç ÷ç ÷çè ø£

- Nodes with label (x1, y1) and (x2, y2) (x1< x2) are connected iff

- Labels are uniformly distributed in the unit square

Limit can be represented as2 2

1 2 2

:[0,1] [0,1] [0,1]1 ,

( , )0,, if

otherwise

Wx x y

W x y

´ ®ì £ïï=íïïî

The limit of randomly grown prefix attachment graphs

(as a function on [0,1]2)

Preferential attachment graph on n fixed nodes

At step m: any two nodes i and j are connected with probability (d(i)+1)(d(j)+1)/(2m+n)2

Allow multiple edges!!!

Repeat until we insert edges. 1

22n

mæö÷ç= ÷ç ÷ç ÷çè ø

A preferential attachment graph

with 200 fixed nodes

and with 5,000 (multiple) edges

( , ) ln( ) ln( )W x z x y=A randomly grown

preferential attachment graph

with 200 fixed nodes ordered by degrees

and with 5,000 (multiple) edges

Proof by computing t(F,Gn)

Can we construct a sequence converging to 1-xy?

Method 1: W-random graph

x1,…,xn,…: independent points from [0,1]

connect xi and xj with probability 1-xi xj

Works for any W

Method 2: growing in order

At step n: - a new node is born, and connected to i with prob (n-i)/n - any two old nodes are connected with probability 1/n

Ignore multiplicity of edges

Works for any monotone decreasing W