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

Physical Review E, Vol. 75, No. 5, 2007

Posted: 26 Nov 2007 Last revised: 9 Apr 2008

See all articles by Maksim Kitsak

Maksim Kitsak

Boston University - Department of Physics

Shlomo Avlin

Bar-Ilan University - Department of Physics

Fabio Pammolli

Polytechnic University of Milan - Department of Management, Economics and Industrial Engineering; CERM Foundation

Massimo Riccaboni

KU Leuven - Department of Managerial Economics, Strategy, and Innovation; IMT Institute for Advanced Studies

H. Eugene Stanley

Boston University - Center for Polymer Studies

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) ∼ C−δ. We find that for non-fractal scale-free networks δ = 2, and for fractal scale-free networks δ = 2 − 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 ℓ∗, separating fractal and non-fractal regimes, scales with dimension dB of the network as p−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.

Suggested Citation

Kitsak, Maksim and Avlin, Shlomo and Pammolli, Fabio and Riccaboni, Massimo and Stanley, H. Eugene, Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks. Physical Review E, Vol. 75, No. 5, 2007, Available at SSRN: https://ssrn.com/abstract=1032610

Maksim Kitsak

Boston University - Department of Physics ( email )

590 Commonwealth Avenue
Boston, MA 02215
United States

Shlomo Avlin

Bar-Ilan University - Department of Physics ( email )

Ramat Gan, 52900
Israel

Fabio Pammolli (Contact Author)

Polytechnic University of Milan - Department of Management, Economics and Industrial Engineering ( email )

Via Lambruschini 4C - building 26/A
Milano, 20156
Italy

CERM Foundation ( email )

Via Fiorentina, 1
Siena, Siena 53100
Italy

HOME PAGE: http://www.cermlab.it

Massimo Riccaboni

KU Leuven - Department of Managerial Economics, Strategy, and Innovation ( email )

Naamsestraat 69 bus 3500
Leuven, 3000
Belgium

IMT Institute for Advanced Studies ( email )

Complesso San Micheletto
Lucca, 55100
Italy

H. Eugene Stanley

Boston University - Center for Polymer Studies ( email )

Boston, MA 02215
United States

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