  Topological data analysis based on persistent homology has been applied to
the molecular dynamics simulation for the fast ion-conducting phase
($\alpha$-phase) of AgI, to show its effectiveness on the ion-migration
mechanism analysis.Time-averaged persistence diagrams of $\alpha$-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 during Ag migration 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.
