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

TL;DR

This paper provides an exact analytical solution for the average distance in a deterministic hierarchical scale-free network, revealing logarithmic growth with network size and comparing it to other network types.

Contribution

It introduces a recursive method to explicitly compute the average distance in a hierarchical scale-free network, demonstrating its logarithmic scaling and relation to other networks.

Findings

Average distance grows logarithmically with network size.

Exact expression for average distance derived and confirmed.

Logarithmic scaling may be a generic feature of deterministic scale-free networks.

Abstract

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.