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 similar to K-gamma with gamma = 1 + ln M/ln(M-1), (ii) the diameter grows logarithmically as D similar to ln N with the number of nodes N, (iii) the clustering coefficient of each node is inversely proportional to its degree, C proportional to 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 similar to B-2. We discuss a classification scheme of scale-free networks into the universality class with the clustering property and the betweenness centrality distribution.