Exact scaling properties of a hierarchical network model

Details Exact scaling properties of a hierarchical network model Exact scaling properties of a hierarchical network model

2002-11-19

arxiv.org/abs/cond-mat/0211399v1

Phys. Rev. E 67, R045103 (2003).

Physics

Condensed Matter

Statistical Mechanics

4 pages

Scientific paper

10.1103/PhysRevE.67.045103

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.

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