Topological data analysis based on persistent homology has been applied to the molecular dynamics simulation for the fast ion-conducting phase (α-phase) of AgI, to show its effectiveness on the ion-migration mechanism analysis. Time-averaged persistence diagrams of α-AgI, which quantitatively records the shape and size of the ring structures in the given atomic configurations, clearly showed the emergence of the four-membered rings formed by two Ag and two I ions at high temperatures. They were identified as common structures during the Ag ion migration. The averaged potential energy change due to the deformation of four-membered ring agrees well with the activation energy calculated from the conductivity Arrhenius plot. The concerted motion of two Ag ions via the four-membered ring was also successfully extracted from molecular dynamics simulations by our approach, providing the new insight into the specific mechanism of the concerted motion.