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.