Network models – random graphs Random networks Random graph

Connectivity of scale-free random graphs
Given:
−γ
N , P(
k)
≈ k for k ≤ κ
Graph properties depend on the degree exponent
• giant cluster:
• connected:
γ ≤ 3.47
γ ≤ 2
W. Aiello, F. Chung, L. Lu, Proc. 32th ACM Theor. Comp., 171 (2000)
M. E. J. Newman, S. H. Strogatz, D. J. Watts, Phys. Rev. E 64, 026118 (2001)
γ
Average path length of scale-free random
graphs
Network:
N , P(
k)
≈ k
ln N + B
Prediction: l sf
= + 1 A , B = f ( γ , κ )
A
A(l-1)-B
15
10
5
food webs
metabolic networks
Internet
collaboration networks
WWW
−γ
0
10 0 10 2 10 4 10 6 10 8 10 10
N
for
k ≤ κ
M. E. J. Newman, S. H. Strogatz, D. J. Watts, Phys. Rev. E 64, 026118 (2001)
• qualitative agreement
• worse than a random
graph
Clustering coefficient of scale-free
random graphs
2
< k > ⎛ < k > − < k >
C =
⎜
2
N ⎝ < k >
2
2
⎞
⎟
⎠
The second term depends on the variance of the degree distribution.
P )
−γ
( k ≈ k
−( 3γ
−7)/(
γ −1)
C ≈ N
For γ < 7/3 C increases with N.
M. E. J. Newman, SIAM Review 45, 167 (2003)
Expectations:
k ≥ 1 giant connected component, k ≥ ln N
γ ≤ 3.47 giant connected component, γ ≤ 2
connected
connected
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