  Nowadays there is a multitude of measures designed to capture different
aspects of network structure. To be able to say if the structure of certain
network is expected or not, one needs a reference model (null model). One
frequently used null model is the ensemble of graphs with the same set of
degrees as the original network. In this paper we argue that this ensemble can
be more than just a null model -- it also carries information about the
original network and factors that affect its evolution. By mapping out this
ensemble in the space of some low-level network structure -- in our case those
measured by the assortativity and clustering coefficients -- one can for
example study how close to the valid region of the parameter space the observed
networks are. Such analysis suggests which quantities are actively optimized
during the evolution of the network. We use four very different biological
networks to exemplify our method. Among other things, we find that high
clustering might be a force in the evolution of protein interaction networks.
We also find that all four networks are conspicuously robust to both random
errors and targeted attacks.
