Degree correlations in Degree correlations in complex networkscomplex networks

Lazaros K. GallosChaoming Song

Hernan A. Makse

Levich Institute, City College of New York

P(k1,k2)Probability that a node with degree k1 is

connected to a node with degree k2.

• Very important but difficult to estimate directly

How we measure correlationsHow we measure correlations

• r : Assortativity coefficient (Newman)

• knn: Average degree of the nearest neighbors (Maslov, Pastor-Satorras)

• ‘Rich-club’ phenomenon (Vespignani)

• : Prob. that two hubs in different boxes are connected (Makse)

Bε

Fractality and renormalizationFractality and renormalization

Song, Havlin, Makse, Nature (2005)Song, Havlin, Makse, Nature Physics (2006)

NN BdB'

kk kdB'

Nodes within a distance belong in the same box

B

WWW

Before… …and after renormalization

ln(h)

Let’s visualize some distributions…Let’s visualize some distributions…

Let’s visualize some distributions…Let’s visualize some distributions…

Before… …and after renormalization

Internet

ln(h)

If If PP((kk11,,kk22) is invariant…) is invariant…

2)1(

121 ),( kkkkP 2121 ),( kkkkP x

Easy to calculate:

Determinescorrelations

)1(111221 ~)(),( kkPkdkkkP

Example: random networksP(k1,k2) = k1P(k1).k2P(k2) = k1

-(-1)k2-(-1)

= -1

How to calculate 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.

bk

bkb

dkkP

dkkkP

kE

)(

)|(

)(22

)(1

1

~)(

kk

kkEb

log P(k)

log kk=10 bk=20

Theory for fractal networksTheory for fractal networks

Bε Prob. that two hubs in different boxes are connected

edBB

~ε

Song et al, Nature Physics (2006)

B

e

k

e

d

d

d

d)1(22

Conservation of links:'2

'1

'2

'12121 ),('')(),( dkdkkkPNdkdkkkNP Bε

Fractals: hub-hub repulsionNon-fractals: hub-hub attraction

In short…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