When transport in networks follows the shortest paths, the union of all shortest path trees $G_{\cup \text{SPT}}$ 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 $G_{\cup \text{SPT}}$. The traffic through the network is examined by the betweenness $B_l$ of links in the overlay $G_{\cup \text{SPT}}$. 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 $\left(\alpha \to 0\right)$, 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 $\text{Pr}\left[B_l=j\right]\sim j^{-c}$. 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.

• Received 31 October 2007

DOI:https://doi.org/10.1103/PhysRevE.77.046105