  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.
