1. Random Graph Models of large networks
Alan Frieze
with special thanks to
Abraham Flaxman

2. Random graph models of large networks - Alan Frieze, CMU
Graphs
A graph G is defined by a vertex set V and an edge set E.
It may look like this:

3. Random graph models of large networks - Alan Frieze, CMU
“Real” Graphs
Large graphs are all around us in the world today, for example, the World Wide Web.
The vertex set V consists of all web pages, and the edge set E consists of all hyperlinks.
|V | ¼ 109, average 7 links per page.

4. Random graph models of large networks - Alan Frieze, CMU
More Examples
Internet
Metabolic Networks
Social Networks
Neural Networks
Peer to Peer Networks
Tunnels of an Ant Colony

5. Random graph models of large networks - Alan Frieze, CMU
Modeling Graphs
We assume that they arise via some random process.
Why not model them as random graphs:
Erdos and Renyi model :
Vertex set {1,2,…,n},
Each of the graphs with
edges equally likely.

6. Random graph models of large networks - Alan Frieze, CMU
Problem
Suppose
For small k the number of vertices degree k is whp
In many real world cases the number of vertices of degree k
for some A, (close to 3 in WWW graph)
e.g. Faloutsos,Faloutsos and Faloutsos.

7. Random graph models of large networks - Alan Frieze, CMU
Fixed Degree Sequence Models:
Choose a degree sequence d1,…,dn and
then choose a graph uniformly from
graphs with this degree sequence.

8. Random graph models of large networks - Alan Frieze, CMU
Molloy and Reed
Let
<0 implies all components of G are small whp.
>0 implies that G contains a giant component (size (n)) whp.

9. Random graph models of large networks - Alan Frieze, CMU
Explanation: represent a vertex of degree k by a set of size k. Randomly pair up points of sets.
Vertex of Degree 3
Expected increase in “free dots” is
idi(di-2)/idi

10. Random graph models of large networks - Alan Frieze, CMU
Aiello, Chung, Lu
Given , they considered random graphs where the number of vertices y of degree x satisfies
For various ranges for ,.

11. Random graph models of large networks - Alan Frieze, CMU
Flavour of results: existence of giant
component whp:
>0= implies there is no giant component
<0 implies there is a unique giant
component.
Bounds given on size of second largest
component too.

12. Cooper and Frieze: Random Digraphs
Random graph models of large networks - Alan Frieze, CMU
Cooper and Frieze: Random Digraphs
They consider random digraphs D with
n vertices where the number of vertices
with in-degree i and out-degree j is li,j .
Let  n=i,jli,j be the number of arcs in D.
Let
d=i,jijli,j/( n).

13. Random graph models of large networks - Alan Frieze, CMU
Flavour of results: size of largest strong
component (size (n)) whp:
d<1 implies there is no giant strong component.
d>1 implies there is a giant strong component S.

14. Random graph models of large networks - Alan Frieze, CMU
More on d>1:
Let L+ be the set of vertices with giant
“fan-out” and L- be the set of vertices
with giant “fan-in”. Then whp
S=L+Å L-

15. Papadimitriou and Mihail:
Random graph models of large networks - Alan Frieze, CMU
Papadimitriou and Mihail:
Model: Fix a1,a2,…an and then add an
edge between vertices i and j with
probability aiaj/An where An= ai.
Let 1¸ 2¸ … be the largest eigenvalues
of the adjacency matrix of the random
graph produced.

16. Random graph models of large networks - Alan Frieze, CMU
Suppose ½<<1. Suppose that ai=a1i-
for small i. Then for small i we have
i» ai where i denotes the i’th
largest degree, and

17. Dynamic models:Preferential Attachment Model (PAM)
Random graph models of large networks - Alan Frieze, CMU
Dynamic models:Preferential Attachment Model (PAM)
We build the graph dynamically:
1. At time t
(a) add vt
(b) connect vt to u chosen randomly
2. Every m steps contract the most recently added m vertices into a single vertex.

18. Random graph models of large networks - Alan Frieze, CMU
1. (b) Connect vt to u chosen randomly
Randomly how?
“The rich get richer”

19. Preferential Attachment Model
Random graph models of large networks - Alan Frieze, CMU
Preferential Attachment Model
History:
Yule, Zoology
Simon, Word frequencies, academic papers, cities, income, more zoology
Barabasi and Albert, WWW

20. Heuristic Analysis of Degrees
Random graph models of large networks - Alan Frieze, CMU
Heuristic Analysis of Degrees
Let dv(t) denote the degree of v at time t. Then,

21. Random graph models of large networks - Alan Frieze, CMU
Suppose that v is added at time s. Then
we get
Thus the number of vertices of degree
exceeding k at time t is

22. Random graph models of large networks - Alan Frieze, CMU
And the number of vertices of degree
exactly k is

23. Random graph models of large networks - Alan Frieze, CMU
A rigorous proof of the following is given
in Bollobas, Riordan, Spencer, Tusnady:
With probablity 1-o(1), as ,
the number of vertices of degree k is

24. Random graph models of large networks - Alan Frieze, CMU
More Work
B. Bollobas, O. Riordan:
Diameter:
Robustness: Suppose we delete the first ct vertices for some c<1.

25. Random graph models of large networks - Alan Frieze, CMU
Researchers have developed similar, but
more complex models which are
mixtures of preferential and random
attachment. These give arbitrary
exponents for the power law.

26. Random graph models of large networks - Alan Frieze, CMU
Copying Model
Communities: A large dense bipartite
sub-graph of the WWW indicates a
“community”. Experiments indicate a
larger number of these than you would
get from say the simple model PAM.
The next model does give many though: it is
due to Kumar,Raghavan,Rajakopalan,Sivakumar
and Upfal.

27. Random graph models of large networks - Alan Frieze, CMU
As in PAM at each stage we add a new vertex vt and we give it m incident edges.
Its construction rests on a parameter .
Then a vertex u is chosen uniformly at random from
Vt={v1,v2,…,vt-1} and then for i=1,2,…,m we
1. With probability  we create edge (vt,x) where x is
chosen randomly from Vt.
2. With probability 1- we create edge (vt,y) where
y was the i’th choice of u.

28. Random graph models of large networks - Alan Frieze, CMU
1. Whp the degree sequence has a power
law with exponent
2. Whp the number of copies of Ki,i, i· m is (te-i).
This contrasts with the simple preferential
model where the expected number is
O(1).

29. Random graph models of large networks - Alan Frieze, CMU
More on PAM
Let Di = i’th max degree at time t:
Fenner,Flaxman,Frieze: Fix k independent
of t.
Whp for any f(t) with f(t)!1 as t!1, and i<k
t1/2/f(t) · i· t1/2f(t)
and
i+1· i - t1/2/f(t).

30. Random graph models of large networks - Alan Frieze, CMU
Furthermore, if 1,…k
are the k’th largest eigenvalues then whp

31. Random graph models of large networks - Alan Frieze, CMU
Crawling on web graphs
Cooper and Frieze considered the
following scenario: We have the model
PAM. Plus there is a spider S which does
a random walk as the graph is growing.
Let m(t) denote the expected number of
vertices not visited at least once by the
spider up to time t.

32. Random graph models of large networks - Alan Frieze, CMU
Let
then

33. Random graph models of large networks - Alan Frieze, CMU
Heuristically Optimized Trade-Offs – Fabrikant, Koutsoupias and Papadimitriou
This a random graph model of the growth
of the internet that exhibits a power law
in its degree sequence.

34. Random graph models of large networks - Alan Frieze, CMU
The model builds a tree on n random
points X1,X2,…,Xn in the unit square [0,1]2
n is a parameter of the model.
Suppose that we have built a tree T on
X1,X2,…,Xi-1 and we wish to connect
Xi up a “close” point on T.

35. Random graph models of large networks - Alan Frieze, CMU
We connect Xi to Xj where j minimises
ndi,j+hj
Here di,j is Euclidean distance and
hj is the tree distance (in edge count)
from Xj to the root X1.

36. Random graph models of large networks - Alan Frieze, CMU
Results
n<2-1/2 implies that T is a star with root X1
n=(n1/2) implies that the degree
distribution of T is exponential.
4· n=o(n1/2) gives a power law
for the degree distribution of T.

37. Random graph models of large networks - Alan Frieze, CMU