Kitsak, Maksim and Havlin, Shlomo and Paul, Gerald and Riccaboni, Massimo and Pammolli, Fabio and Stanley, H. Eugene
Betweenness centrality of fractal and nonfractal scale-free model networks and tests on real networks.
Physical Review E, 75 (5).
056115.
ISSN 1539-3755
(2007)
Abstract
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)∼C−δ. We find that for nonfractal scale-free networks δ=2, and for fractal scale-free networks δ=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.
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