Exact scaling properties of a hierarchical network model

TL;DR

This paper derives exact scaling properties of a hierarchical network model, revealing power-law distributions and logarithmic growth in key network metrics, with implications for classifying scale-free networks.

Contribution

It provides exact analytical results for degree, diameter, clustering, and betweenness centrality in a hierarchical network model, advancing understanding of their scaling behaviors.

Findings

Degree distribution follows a power law with exponent depending on M

Diameter grows logarithmically with network size

Clustering coefficient inversely proportional to degree

Abstract

We report on 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 $P_K \sim K^{-\gamma}$ with $\gamma = 1+\ln M /\ln (M-1)$, (ii) the diameter grows logarithmically as $D \sim \ln N$ with the number of nodes $N$, (iii) the clustering coefficient of each node is inversely proportional to its degree, $C \propto 1/K$, and the average clustering coefficient is nonzero in the infinite $N$ limit, and (iv) the betweenness centrality distribution follows a power law $P_B \sim B^{-2}$. We discuss a classification scheme of scale-free networks into the universality class with the clustering property and the betweenness centrality distribution.