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

摘要

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