Betweenness centrality of fractal and nonfractal scale-free model networks and tests on real networks

Maksim Kitsak, Shlomo Havlin, Gerald Paul, Massimo Riccaboni, Fabio Pammolli, and H. Eugene Stanley
Phys. Rev. E 75, 056115 – Published 31 May 2007

Abstract

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)Cδ. We find that for nonfractal scale-free networks δ=2, and for fractal scale-free networks δ=21dB, 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 *, separating fractal and nonfractal regimes, scales with dimension dB of the network as p1dB, 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.

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  • Received 8 November 2006

DOI:https://doi.org/10.1103/PhysRevE.75.056115

©2007 American Physical Society

Authors & Affiliations

Maksim Kitsak1, Shlomo Havlin1,2, Gerald Paul1, Massimo Riccaboni3, Fabio Pammolli1,3,4, and H. Eugene Stanley1

  • 1Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
  • 2Minerva Center and Department of Physics, Bar-Ilan University, Ramat Gan, Israel
  • 3Faculty of Economics, University of Florence, Via delle Pandette 9, Florence, 50127, Italy
  • 4IMT Institute for Advanced Studies, Via S. Micheletto 3, Lucca, 55100, Italy

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Issue

Vol. 75, Iss. 5 — May 2007

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