Betweenness centrality in a weighted network.

When transport in networks follows the shortest paths, the union of all shortest path trees G union or logical sum SPT can be regarded as the "transport overlay network." Overlay networks such as peer-to-peer networks or virtual private networks can be considered as a subgraph of G union or logical sum SPT. The traffic through the network is examined by the betweenness Bl of links in the overlay G union or logical sum SPT. The strength of disorder can be controlled by, e.g., tuning the extreme value index alpha of the independent and identically distributed polynomial link weights. In the strong disorder limit (alpha-->0), all transport flows over a critical backbone, the minimum spanning tree (MST). We investigate the betweenness distributions of wide classes of trees, such as the MST of those well-known network models and of various real-world complex networks. All these trees with different degree distributions (e.g., uniform, exponential, or power law) are found to possess a power law betweenness distribution Pr[Bl=j] approximately j(-c). The exponent c seems to be positively correlated with the degree variance of the tree and to be insensitive of the size N of a network. In the weak disorder regime, transport in the network traverses many links. We show that a link with smaller link weight tends to carry more traffic. This negative correlation between link weight and betweenness depends on alpha and the structure of the underlying topology.