Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the in- teger Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations, leading to fluctuations within the network. We have developed a mean field Virtual Pebble Game (VPG) that provides a probabilistic description of the interaction network, meaning that sampling is not required. We extensively test the VPG algorithm over a variety of body-bar networks created on disordered lattices, from these calculations we fully characterize the network conditions under which the performance of the VPG offers the best solution. The VPG provides a satisfactory description of the ensemble averaged PG properties, especially in regions removed from the rigidity transition where ensemble fluctuations are greatest. In further experiments, we characterized the VPG across a structurally nonredundant dataset of 272 proteins. Using quantitative and visual assessments of the rigidity characterizations, the VPG results are shown to accurately reflect the ensemble averaged PG properties. That is, the fluctuating interaction network is well represented by a single calculation that re- places density functions with average values, thus speeding up the desired calculation by several orders of magnitude. Finally, we propose a new algorithm that is based on the combination of PG and VPG to balance the amount of sampling and mean field treatment. While offering interesting results, this approach needs to be further optimized to fully lever- age its utility. All these results positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability
