  A probabilistic generative network model with $n$ nodes and $m$ overlapping
layers is obtained as a superposition of $m$ mutually independent Bernoulli
random graphs of varying size and strength. When $n$ and $m$ are large and of
the same order of magnitude, the model admits a sparse limiting regime with a
tunable power-law degree distribution and nonvanishing clustering coefficient.
In this article we prove an asymptotic formula for the joint degree
distribution of adjacent nodes. This yields a simple analytical formula for the
model assortativity, and opens up ways to analyze rank correlation coefficients
suitable for random graphs with heavy-tailed degree distributions. We also
study the effects of power laws on the asymptotic joint degree distributions.
