Transport Properties of Fractal and Non-Fractal Scale-Free Networks

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), 2007. http://arxiv.org/abs/physics/0702001

2. Motivation and Objectives Betweenness Centrality C L.C. Freeman, 1979 (Approximately equal to the number of paths passing specific node) How do we identify central nodes? • 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. • 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? Small degree nodes in fractal network may have large centrality!!! 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)

4. Centrality vs Degree Correlation Rewire many times Preserve degrees • One can’t compare centralities of networks directly due to uniqueness of real networks. • The network can be compared to its random counterpart ! • 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 Dispersion Correlation Coefficient: Fractal SF networks Non-Fractal SF networks Weak centrality vs degree correlation results in high variance of centrality values attributed to nodes of given degree

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