Degree correlations in scale-free null models

TL;DR

This paper investigates how the average nearest neighbor degree $a(k)$ behaves in large-scale power-law networks, showing it decays as a power law in three null models, reflecting a common structural property.

Contribution

It demonstrates that in three simple null models, the average nearest neighbor degree decays as a power law, matching empirical observations in real-world networks.

Findings

$a(k)$ decays beyond $n^{( au-2)/( au-1)}$

$a(k)$ settles on a power law $k^{ au-3}$

Decay behavior is consistent across all three 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.