Limits of randomly grown
graph sequences
Katalin Vesztergombi
Eötvös University, Budapest
With: Christian Borgs, Jennifer Chayes,
László Lovász, Vera Sós
Convergent graph sequences
hom( F , G)
t ( F , G) 
| V (G) ||V ( F )|
Probability that random map
V(F)V(G) is a hom
(G1 , G2 ,...) convergent: " F t ( F , Gn ) isconvergent
Example: random graphs
t ( F , G(n,
1
2 )) ®
| E ( F )|
()
1
2
with probability 1
The limit object
W0 = {W : [0,1]2 ® [0,1] symmetric, measurable}
t ( F ,W ) =
ò Õ
W ( xi , x j ) dx
[0,1]V ( F ) ij Î E ( F )
Gn ® W :
" F t ( F , Gn ) ® t ( F ,W )
For every convergent graph sequence (Gn)
there is a W Î W0 such that Gn ® W
Lovász-Szegedy
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
Half-graphs
A random graph with
100 nodes and with edge density 1/2
W1/2
Rearranging the rows and columns
A random graph with
100 nodes and with edge density 1/2
(no matter how you reorder the nodes)
W1/2
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
After n steps:
probability that nodes i < j are not connected:
j- 1 j
n- 1 j - 1
L
=
j j+ 1
n
n
expected degree of j:
n - 1 ( j - 1)( j - 2)
2
2n
expected number of edges:
n2 - 1
6
The limit:
probability that nodes i and j are connected:
max(i, j ) - 1
1n
if i=xn and j=yn
» 1- max( x, y )
These are independent events for different i,j.
A randomly grown
uniform attachment graph
W ( x, y ) = 1- max( x, y )
with 200 nodes
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
This tends to some
shades of gray; is that
the limit?
No, by computing
triangle densities!
Is this graph sequence
convergent at all?
A randomly grown prefix attachment graph
with 200 nodes
Yes, by computing
subgraph densities!
This also tends to some
shades of gray; is that
the limit?
No…
A randomly grown prefix attachment graph
with 200 nodes (ordered by degrees)
Label node born in step k, connecting to {1,…,m},
by (k/n, m/k)
- Labels are uniformly distributed in the unit square
- Nodes with label (x1, y1) and (x2, y2) (x1< x2) are
connected iff
x1 £ x2 y2
æk1 k2 m2
çç £
çè n
n k2
Limit can be represented as
W :[0,1]2 ´ [0,1]2 ® [0,1]
ìïï 1, if x1 £ x2 y2 ,
W ( x, y ) = í
ïïî 0, otherwise
or
ö
÷
k1 £ m2 ÷
÷
÷
ø
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!!!
nö
1æ
÷ edges.
ç
Repeat until we insert m = ç ÷
÷
÷
2 çè2ø
A preferential attachment graph
with 200 fixed nodes
and with 5,000 (multiple) edges
Proof by computing
t(F,Gn)
A randomly grown
preferential attachment graph
with 200 fixed nodes ordered by degrees
and with 5,000 (multiple) edges
W ( x, z ) = ln( x) ln( y)
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