Betweenness centrality in a weighted network

When transport in networks follows the shortest paths, the union of all shortest path trees G_∪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_∪SPT . The traffic through the network is examined by the betweenness B_l of links in the overlay G_∪SPT . The strength of disorder can be controlled by, e.g., tuning the extreme value index α of the independent and identically distributed polynomial link weights. In the strong disorder limit (α→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[B_l=j]∼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 α and the structure of the underlying topology.