Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks

We study the betweenness centrality of fractal and non-fractal 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 non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution P(C) amp;#8764; Camp;#8722;amp;#948;. We find that for non-fractal scale-free networks amp;#948; = 2, and for fractal scale-free networks amp;#948; = 2 amp;#8722; 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 AS 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 non-fractal networks upon adding random edges to a fractal network. We show that the crossover length amp;#8467;amp;#8727;, separating fractal and non-fractal regimes, scales with dimension dB of the network as pamp;#8722;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