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

Abstract

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 ${\ell }^{\ast }$, 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 $p$.

arXiv comment19 pages, 6 figures. Submitted to PRE

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