1. Transport Properties of Fractal and Non-Fractal Scale-Free Networks
M. Kitsak, S. Havlin, G. Paul, M. Riccaboni, F. Pammolli and H.E. Stanley
M. Kitsak et al, Phys Rev E, (in press),

2. Betweenness Centrality C
Motivation and Objectives
Most of the transport on the network flows along the shortest paths.
Central nodes/edges are critical: if they are blocked – transport becomes inefficient.
It’s important to identify and “improve” central nodes/edges.
How do we identify central nodes?
Betweenness Centrality C
L.C. Freeman, 1979
(Approximately equal to the number of paths passing specific node)
Do nodes of high centrality have large degree? (in scale – free networks)
What is an appropriate measure of correlation between centrality and degree?

3. Why Fractal Networks?
Many real networks are scale-free (SF) fractals. (WWW, Biological… )
The key principle of SF fractal networks is “Repulsion Between Hubs”:
Large degree nodes in SF fractal networks tend to connect to small
degree nodes and not to each other.
C. Song et al, Nature Physics 2, 275, (2006)
Non-fractal (hubs connect directly)
Fractal (hubs connect to small degree nodes)
Small degree nodes in fractal network may have large centrality!!!

4. The network can be compared to its random counterpart !
Centrality vs Degree Correlation
One can’t compare centralities of networks directly due to uniqueness of real networks.
The network can be compared to its random counterpart !
Rewire many times
Preserve degrees
Rewired network has degree distribution identical to the original network.
Repulsion between hubs is broken by random rewiring. The random network is always non-fractal.

5. Centrality vs Degree Correlation
Weak centrality vs degree correlation results in high variance
of centrality values attributed to nodes of given degree
Centrality vs degree correlation of small degree nodes in non-fractal scale-free networks is much stronger than that in scale-free fractal networks.

6. Dispersion Correlation Coefficient
Weak centrality vs degree correlation results in high variance of centrality values attributed to nodes of given degree
Dispersion Correlation Coefficient:
Fractal SF networks
Non-Fractal SF networks

7. Summary and Conclusions
1. Centrality vs degree correlation is much weaker in fractal networks than in non-fractal.
Fractal networks should be more stable to intentional attacks.
Immunization/Attack strategies should be optimized for fractal networks.
Dispersion Correlation Coefficient R allows to distinguish between fractal and non-fractal networks.
Power-law centrality distribution
Centralities of nodes are larger in fractal scale-free networks. Fractal networks have transport properties different from non-fractal.
4. Transition from fractal to non-fractal networks occurs upon adding random edges to a fractal network.
A crossover is observed from fractal to non-fractal networks.
Relatively small percent of edges is needed to turn fractal network into non-fractal.
3. and 4. were not discussed