This preview shows pages 1–3. Sign up to view the full content.
1
Properties common to many largescale networks, independently of
their origin and function:
1. The degree and betweenness distribution are decreasing
functions, usually powerlaws.
2. The distances scale logarithmically with the network size
Network models – random graphs
N
log
3. The clustering coefficient does not seem to depend on the
network size
As though all these networks were part of the same family/class.
k
log
l
≈
k
C
∝
The average distance and clustering coefficient only depend on the
number of nodes and edges in the network.
This suggests that general models based only on the number of
nodes and edges in the network could be successful in describing
the properties of an “expected” (characteristic) network.
Random networks
Uniformly random network: distributes the edges uniformly among
nodes.
Probabilistic interpretation:
There exists a set (ensemble) of networks with given number of
nodes and edges. Select a random member of this set.
What are the expected properties of this network? – studied by
random graph theory
.
Ex. 1
Start with 10 isolated nodes. For each pair of nodes, throw with a dice,
and connect them if the number on the dice is
1
. Describe the graph you
obtained. How many edges are in the graph? Is it connected or not? What
is the average degree and the degree distribution?
E2
Ex. 2
Now connect node pairs if the number on the dice is
1 or 2.
How is the
graph different from the previous case?
Ex. 2
How many edges do you expect a graph with N nodes would have if each
edge is selected with throwing with a dice?
Random graph theory
ErdösRényi algorithm 
Publ. Math. Debrecen 6, 290 (1959)
• fixed node number
N
• connecting pairs of nodes
with probability
p
Random graph theory studies the expected properties of graphs with
2
)
1
N
(
N
p
E
−
=
Expected number of edges:
∞
→
N
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
The properties of random graphs depend
on
p
Properties studied:
is the graph connected?
does the graph contain a giant connected component?
what is the diameter of the graph?
does the graph contain cliques (complete subgraphs)?
Probabilistic formulation: what is the probability that a graph with
N
nodes and connection probability
p
is connected?
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09
 Power, Work

Click to edit the document details