Transcript Degree correlations in complex networks Lazaros K. Gallos Chaoming Song Hernan A. Makse Levich Institute, City College of New York.

Degree correlations in
complex networks
Lazaros K. Gallos
Chaoming Song
Hernan A. Makse
Levich Institute, City College of New York
Probability that a node with degree k1 is
connected to a node with degree k2.
P(k1,k2)
• Very important but difficult to estimate directly
How we measure correlations
• r : Assortativity coefficient (Newman)
• knn: Average degree of the nearest
neighbors (Maslov, Pastor-Satorras)
• ‘Rich-club’ phenomenon (Vespignani)
•  B  : Prob. that two hubs in different
boxes are connected (Makse)
ε
Fractality and renormalization
Nodes within a distance  B
belong in the same box
N '   Bd B N
k '  Bdk k
Song, Havlin, Makse, Nature (2005)
Song, Havlin, Makse, Nature Physics (2006)
Let’s visualize some distributions…
WWW
ln(h)
Before…
…and after renormalization
Let’s visualize some distributions…
Internet
ln(h)
Before…
…and after renormalization
If P(k1,k2) is invariant…
Easy to calculate:
 (  1)
P
(
k
,
k
)
dk

k
P
(
k
)
~
k
 1 2 2 1 1 1

(x1) 
11
2 2
PP((kk11, ,kk22))kk k k
Determines
correlations
Example: random networks
P(k1,k2) = k1P(k1).k2P(k2) = k1-(-1)k2-(-1)
 =  -1
How to calculate 
We define the quantity Eb(k) as the prob. that a node
with degree k is connected to nodes with degree
larger than bk.

log P(k)
 P(k | k )dk
2
Eb (k )  bk
2

 P(k )dk
bk
k=10
1
k
Eb (k ) ~ 1  k (  )
k
bk=20
log k
Theory for fractal networks
ε 
B
Prob. that two hubs in different boxes are connected
ε
B
 ~ B
 de
Fractals: hub-hub repulsion
Non-fractals: hub-hub attraction
Song et al, Nature Physics (2006)
Conservation of links:
ε
NP(k1, k2 )dk1dk2   ( B ) N ' P' (k1' , k2' )dk1' dk2'
de
de
  2   2  (  1)
dk
dB
In short…
• The joint degree distribution P(k1,k2) can
be described with one unique exponent .
• Networks with different correlation
properties are clustered in different areas
of the (,) space