Betweenness centrality of fractal and nonfractal scale-free model networks and tests on real networks.
We study the betweenness centrality of fractal and nonfractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality C of nodes is much weaker in fractal network models compared to nonfractal models. We also show that nodes of both fractal and nonfractal scale-free networks have power-law betweenness centrality distribution P(C) approximately C(-delta). We find that for nonfractal scale-free networks delta=2, and for fractal scale-free networks delta=2-1/dB, where dB is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at the autonomous system level (N=20566), where N is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to nonfractal networks upon adding random edges to a fractal network. We show that the crossover length l*, separating fractal and nonfractal regimes, scales with dimension dB of the network as p(-1/dB), where p is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with p.
Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA.
Maksim
M
Kitsak
Shlomo
S
Havlin
Gerald
G
Paul
Massimo
M
Riccaboni
Fabio
F
Pammolli
H Eugene
HE
Stanley