  The past decade has witnessed the development and success of coarse-grained
network models of proteins for predicting many equilibrium properties related
to collective modes of motion. Curiously, the results are usually robust
towards the different cutoff distances used for constructing the residue
networks from the knowledge of the experimental coordinates. In this study, we
present a systematical study of network construction, and their effect on the
predicted properties. Probing bond orientational order around each residue, we
propose a natural partitioning of the interactions into an essential and a
residual set. In this picture the robustness originates from the way with which
new contacts are added so that an unusual local orientational order builds up.
These residual interactions have a vanishingly small effect on the force
vectors on each residue. The stability of the overall force balance then
translates into the Hessian as small shifts in the slow modes of motion and an
invariance of the corresponding eigenvectors. We introduce a rescaled version
of the Hessian matrix and point out a link between the matrix Frobenius norm
based on spectral stability arguments. A recipe for the optimal choice of
partitioning the interactions into essential and residual components is
prescribed. Implications for the study of biologically relevant properties of
proteins are discussed with specific examples.
