Exact scaling properties of a hierarchical network model

We report on the exact results for the degree K, the diameter D, the clustering coefficient C, and the betweenness centrality B of a hierarchical network model with a replication factor M. Such quantities are calculated exactly with the help of recursion relations. Using the results, we show that (i) the degree distribution follows a power law $P_K\sim K^{-\gamma }$ with $\gamma =1+\ln M/\ln \left(M-1\right),$ (ii) the diameter grows logarithmically as $D\sim \ln N$ with the number of nodes N, (iii) the clustering coefficient of each node is inversely proportional to its degree, $C\propto 1/K,$ and the average clustering coefficient is nonzero in the infinite N limit, and (iv) the betweenness centrality distribution follows a power law $P_B\sim B^{-2}.$ We discuss a classification scheme of scale-free networks into the universality class with the clustering property and the betweenness centrality distribution.