  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 $P(C)\sim C^{-\delta}$. We find that for non-fractal
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 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 $\ell^{*}$, separating fractal and
non-fractal 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 $p$.
