Betweenness centrality in large complex networks. Barthélemy, M. *The European Physical Journal B*, 38(2):163–168, March, 2004. Paper doi abstract bibtex .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 ηη\eta. We find that for trees or networks with a small loop density η=2η=2\eta = 2 while a larger density of loops leads to η\textless2η\textless2\eta \textless 2. For scale-free networks characterized by an exponent γγ\gamma which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent δδ\delta. We show that this exponent δδ\delta must satisfy the exact bound δ≥(γ+1)/2δ≥(γ+1)/2\delta\geq (\gamma + 1)/2. If the scale free network is a tree, then we have the equality δ=(γ+1)/2δ=(γ+1)/2\delta = (\gamma + 1)/2.

@article{barthelemy_betweenness_2004,
title = {Betweenness centrality in large complex networks},
volume = {38},
issn = {1434-6028, 1434-6036},
url = {http://link.springer.com/article/10.1140/epjb/e2004-00111-4},
doi = {10.1140/epjb/e2004-00111-4},
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 ηη{\textbackslash}eta. We find that for trees or networks with a small loop density η=2η=2{\textbackslash}eta = 2 while a larger density of loops leads to η{\textless}2η{\textless}2{\textbackslash}eta {\textless} 2. For scale-free networks characterized by an exponent γγ{\textbackslash}gamma which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent δδ{\textbackslash}delta. We show that this exponent δδ{\textbackslash}delta must satisfy the exact bound δ≥(γ+1)/2δ≥(γ+1)/2{\textbackslash}delta{\textbackslash}geq ({\textbackslash}gamma + 1)/2. If the scale free network is a tree, then we have the equality δ=(γ+1)/2δ=(γ+1)/2{\textbackslash}delta = ({\textbackslash}gamma + 1)/2.},
language = {en},
number = {2},
urldate = {2016-12-15},
journal = {The European Physical Journal B},
author = {Barthélemy, M.},
month = mar,
year = {2004},
pages = {163--168},
}

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