Network models – random graphs Random networks Random graph

Are real networks like random graphs?
As quantitative data about real networks becomes available, we can
compare their topology with that of random graphs.
Starting measures: N, for the real network.
Determine l, C and P(k) for a random graph with the same N and .
log N
l rand
≈
log k
P
C rand
= p =
k k
N −1−k
rand
( k ) ≅ C
N −1
p ( 1 − p )
k
N
l log
15
10
5
Path length and order in real networks
log N
k
l rand
=
C
log k
rand
=
N
food webs
neural network
power grid
collaboration networks
WWW
metabolic networks
Internet
C/
10 0
10 -2
food webs
10 -4 neural network
metabolic networks
power grid
collaboration networks
10 -6 WWW
Measure l, C and P(k) for the real network. Compare.
0
10 0 10 2 10 4 10 6 10 8 10 10
N
10 -8
10 0 10 2 10 4 10 6 10 8
N
Real networks have short distances like random graphs yet show
signs of local order.
The degree distribution of the WWW is a
power-law
Power-law degree distributions were found in
diverse networks
Internet, router level
Actor collaboration
P
out
( k)
≈ k
P ( k)
≈ k
in
−2.
45
−2.
1
P(
k)
≈ k
P(k)
−2.4
P(
k)
≈ k
−2.3
R. Albert, H. Jeong, A.-L. Barabási, Nature 401, 130 (1999)
A. Broder et al., Comput. Netw. 33, 309 (1999)
0 1 2 3
1 2 3
10 10 10 10 10 10 10
k
R. Govindan, H. Tangmunarunkit, IEEE Infocom (2000)
A.-L. Barabási, R. Albert, Science 286, 509 (1999)
4