Amino acid networks (AANs) abstract the protein structure by recording the amino acid contacts and can provide insight into protein function. Herein, we describe a novel AAN construction technique that employs the rigidity analysis tool, FIRST, to build the AAN, which we refer to as the residue geometry network (RGN). We show that this new construction can be combined with network theory methods to include the effects of allowed conformal motions and local chemical environments. Importantly, this is done without costly molecular dynamics simulations required by other AAN-related methods, which allows us to analyse large proteins and/or data sets. We have calculated the centrality of the residues belonging to 795 proteins. The results display a strong, negative correlation between residue centrality and the evolutionary rate. Furthermore, among residues with high closeness, those with low degree were particularly strongly conserved. Random walk simulations using the RGN were also successful in identifying allosteric residues in proteins involved in GPCR signalling. The dynamic function of these residues largely remain hidden in the traditional distance-cutoff construction technique. Despite being constructed from only the crystal structure, the results in this paper suggests that the RGN can identify residues that fulfil a dynamical function.