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 $P\left(C\right)\sim C^{-\delta }$. We find that for nonfractal scale-free networks $\delta =2$, and for fractal scale-free networks $\delta =2-1∕d_B$, where $d_B$ 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 ${\ell }^{*}$, separating fractal and nonfractal regimes, scales with dimension $d_B$ of the network as $p^{-1∕d_B}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.