Exploring the assortativity-clustering space of a network's degree sequence

Nowadays there is a multitude of measures designed to capture different aspects of network structure. To be able to say if a measured value is expected or not, one needs to compare it with 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. Here, we argue that this ensemble can give more information about the original network than effective values of network structural quantities. 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 where in the valid region of the parameter space the observed networks are. Such analysis suggests which quantities (or combination of 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.