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 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 ≥ (γ + 1)/2. If the scale free network is a tree, then we have the equality δ = (γ + 1)/2. PACS. 89.75.-k Complex systems – 89.75.Hc Networks and genealogical trees – 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion