Macroscopic elastic properties of materials depend on the underlying microscopic structures. We have investigated the topological structure of three-dimensional network glass, such as vitreous SiO2, and its effect on the rigidity, using a parallel molecular-dynamics (MD) approach. The topological analysis based on the graph theory is employed to characterize disordered networks in the computer generated model of vitreous SiO2. The nature of connectivity of the elementary units beyond the nearest-neighbor, which is related to the medium-range order structure of amorphous state, is described in terms of the ring distribution by the shortest-path analysis. In large-scale MD simulations, the task of detecting these rings from a large amount of data is computationally demanding. Elastic moduli of vitreous SiO2 are calculated with the fluctuation formula for internal stress. The quantitative relation between the statistics of rings for vitreous SiO2 and the elastic moduli are discussed.