Degree correlations in scale-free null models

Abstract

We study the average nearest neighbor degree $a(k)$ of vertices with degree $k$. In many real-world networks with power-law degree distribution $a(k)$ falls off in $k$, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that $a(k)$ indeed decays in $k$ in three simple random graph null models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph and the hyperbolic random graph. We consider the large-network limit when the number of nodes $n$ tends to infinity. We find for all three null models that $a(k)$ starts to decay beyond $n^{(\tau -2)/(\tau -1)}$ and then settles on a power law $a(k)\sim k^{\tau -3}$, with $\tau$ the degree exponent.

arXiv comment21 pages, 4 figures

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Cited by in corpus (2)

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