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 $η$. We find that for trees or networks with a small loop density $η=2$ while a larger density of loops leads to $η<2$. For scale-free networks characterized by an exponent $γ$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $δ$. We show that this exponent $δ$ must satisfy the exact bound $δ\geq (γ+1)/2$. If the scale free network is a tree, then we have the equality $δ=(γ+1)/2$.