Betweenness centrality in large complex networks

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 greater than or equal to (gamma + 1)/2. If the scale free network is a tree, then we have the equality delta = (gamma + 1)/2.