  Correlations may affect propagation processes on complex networks. To analyze
their effect, it is useful to build ensembles of networks constrained to have a
given value of a structural measure, such as the degree-degree correlation $r$,
being random in other aspects and preserving the degree distribution. This can
be done through Monte Carlo optimization procedures. Meanwhile, when tuning
$r$, other network properties may concomitantly change. Then, in this work we
analyze, for the $r$-ensembles, the impact of $r$ on properties such as
transitivity, branching and characteristic lengths, that are relevant when
investigating spreading phenomena on these networks. The present analysis is
performed for networks with degree distributions of two main types: either
localized around a typical degree (with exponentially bounded asymptotic decay)
or broadly distributed (with power-law decay). Correlation bounds and size
effects are also investigated.
