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

2. P(k1,k2)P(k1,k2) Probability that a node with degree k 1 is connected to a node with degree k 2. Very important but difficult to estimate directly

3. How we measure correlations r : Assortativity coefficient (Newman) k nn : Average degree of the nearest neighbors (Maslov, Pastor-Satorras) ‘Rich-club’ phenomenon (Vespignani) : Prob. that two hubs in different boxes are connected (Makse)

4. Fractality and renormalization Song, Havlin, Makse, Nature (2005) Song, Havlin, Makse, Nature Physics (2006) Nodes within a distance belong in the same box

5. WWW Before……and after renormalization ln(h) Let’s visualize some distributions…

6. Before……and after renormalization Internet ln(h)

7. If P(k 1,k 2 ) is invariant… Easy to calculate: Determines correlations Example: random networks P(k 1,k 2 ) = k 1 P(k 1 ). k 2 P(k 2 ) = k 1 -(  -1) k 2 -(  -1)  =  -1

8. How to calculate  We define the quantity E b (k) as the prob. that a node with degree k is connected to nodes with degree larger than bk. log P(k) log k k=10 bk=20

11. Theory for fractal networks Prob. that two hubs in different boxes are connected Song et al, Nature Physics (2006) Conservation of links: Fractals: hub-hub repulsion Non-fractals: hub-hub attraction

13. In short… The joint degree distribution P(k 1,k 2 ) can be described with one unique exponent . Networks with different correlation properties are clustered in different areas of the ( ,  ) space