  Various real systems simultaneously exhibit scale-free and hierarchical
structure. In this paper, we study analytically average distance in a
deterministic scale-free network with hierarchical organization. Using a
recursive method based on the network construction, we determine explicitly the
average distance, obtaining an exact expression for it, which is confirmed by
extensive numerical calculations. The obtained rigorous solution shows that the
average distance grows logarithmically with the network order (number of nodes
in the network). We exhibit the similarity and dissimilarity in average
distance between the network under consideration and some previously studied
networks, including random networks and other deterministic networks. On the
basis of the comparison, we argue that the logarithmic scaling of average
distance with network order could be a generic feature of deterministic
scale-free networks.
