Average distance in a hierarchical scale-free network: an exact solution

Abstract

Various real systems simultaneously exhibit scale-free and hierarchical structure. In thispaper, we study analytically average distance in a deterministic scale-free network withhierarchical organization. Using a recursive method based on the network construction, wedetermine explicitly the average distance, obtaining an exact expression for it, which isconfirmed by extensive numerical calculations. The rigorous solution obtainedshows that the average distance grows logarithmically with the network order(number of nodes in the network). We show the similarity and dissimilarity inaverage distance between the network under consideration and some previouslystudied networks, including random networks and other deterministic networks. Onthe basis of the comparison, we argue that the logarithmic scaling of averagedistance with network order could be a generic feature of deterministic scale-freenetworks.