Betweenness Centrality in Large Complex Networks

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 $\delta \ge (\gamma +1)/2$. If the scale free network is a tree, then we have the equality $\delta =(\gamma +1)/2$.

6 pages, 5 figures, revised version