Abstract.
We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent \(\eta\). We find that for trees or networks with a small loop density \(\eta = 2\) while a larger density of loops leads to \(\eta < 2\). For scale-free networks characterized by an exponent \(\gamma\) which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent \(\delta\). We show that this exponent \(\delta\) must satisfy the exact bound \(\delta\geq (\gamma + 1)/2\). If the scale free network is a tree, then we have the equality \(\delta = (\gamma + 1)/2\).
Similar content being viewed by others
References
C. Bergé, Graphs and Hypergraphs, 2nd edn. (North-Holland, Amsterdam, 1976)
J. Clark, D.A. Holton, A first look at graph theory (World Scientific, 1991)
R. Albert, H. Jeong, A.-L. Barabási, Nature (London) 401, 130 (1999)
R. Cohen, K. Erez, D. benAvraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)
P. Holme, B.J. Kim, C.N. Yoon, S.K. Han, Phys. Rev. E 65, 056109 (2002)
R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)
L.C. Freeman, Sociometry 40, 35 (1977)
S. Wasserman, K. Faust, Social Network Analysis: Methods and applications (Cambridge University Press, 1994)
U. Brandes, J. Math. Soc. 25, 163 (2001)
D. Wilkinson, B.A. Huberman, cond-mat/0210147
L.C. Freeman, S.P. Borgatti, D.R. White, Social Networks 13, 141 (1991)
M.E.J. Newman, cond-mat/0309045
L.A.N. Amaral, A. Scala, M. Barthélemy, H.E. Stanley, Proc. Natl. Acad. Sci. USA 97, 11149 (2000)
K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 87, 278701 (2001)
A.-L. Barabasi, R. Albert, Science 286, 509 (1999)
For networks with peaked connectivity distributions such as the random graph, the centrality is also peaked and the exponent \(\delta\) is not defined
K.-I. Goh, H. Jeong, B. Kahng, D. Kim, Proc. Natl. Acad. Sci. (USA) 99, 12583 (2002)
L.P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization (World Scientific, 2000)
M. Barthélemy, Phys. Rev. Lett. 91, 189803 (2003)
A. Vazquez, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 066130 (2002)
S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)
K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 91, 189804 (2003)
G. Szabo, M. Alava, J. Kertesz, Phys. Rev. E 66, 036101 (2002)
B. Bollobas, Random Graph (Academic Press, New York, 1985)
A. Renyi, Probability theory (New York, Elsevier, 1980)
D.J. Watts, D.H. Strogatz, Nature 393, 440 (1998)
It would be interesting to quantify for different types of networks the degree of anisotropy--measured by the N i ‘s--versus the connectivity
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: 11 December 2003, Published online: 14 May 2004
PACS:
89.75.-k Complex systems - 89.75.Hc Networks and genealogical trees - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
Rights and permissions
About this article
Cite this article
Barthélemy, M. Betweenness centrality in large complex networks. Eur. Phys. J. B 38, 163–168 (2004). https://doi.org/10.1140/epjb/e2004-00111-4
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2004-00111-4