  An important problem in modeling networks is how to generate a randomly
sampled graph with given degrees. A popular model is the configuration model, a
network with assigned degrees and random connections. The erased configuration
model is obtained when self-loops and multiple edges in the configuration model
are removed. We prove an upper bound for the number of such erased edges for
regularly-varying degree distributions with infinite variance, and use this
result to prove central limit theorems for Pearson's correlation coefficient
and the clustering coefficient in the erased configuration model. Our results
explain the structural correlations in the erased configuration model and show
that removing edges leads to different scaling of the clustering coefficient.
We then prove that for the rank-1 inhomogeneous random graph, another null
model that creates scale-free simple networks, the results for Pearson's
correlation coefficient as well as for the clustering coefficient are similar
to the results for the erased configuration model.
