Degree correlations in complex networks

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Degree correlations in complex networks

• Lazaros K. Gallos
• Chaoming Song
• Hernan A. Makse
• Levich Institute, City College of New York

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• Probability that a node with degree k1 is
  connected to a node with degree k2.
• Very important but difficult to estimate directly

P(k1,k2)
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How 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)

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Fractality and renormalization
Song, Havlin, Makse, Nature (2005) Song, Havlin,
Makse, Nature Physics (2006)
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Lets visualize some distributions
WWW
ln(h)
Before
and after renormalization
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Lets visualize some distributions
Internet
ln(h)
Before
and after renormalization
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If P(k1,k2) is invariant
Easy to calculate
Determines correlations
Example random networks P(k1,k2)
k1P(k1).k2P(k2) k1-(g-1)k2-(g-1) e g -1
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How to calculate e

• 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)
log k
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Theory for fractal networks
Prob. that two hubs in different boxes are
connected
Fractals hub-hub repulsion Non-fractals hub-hub
attraction
Song et al, Nature Physics (2006)
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In short

• The joint degree distribution P(k1,k2) can be
  described with one unique exponent e.
• Networks with different correlation properties
  are clustered in different areas of the (g,e)
  space