In this talk, I present a new method to detect robust common topological structures of two geometric objects. The idea is to extend the notion of persistent homology to representations on a commutative triple ladder quiver. (i) I show that representations on the commutative triple ladder quiver are finite type. (ii) The Auslander-Reiten quiver of the commutative triple ladder, which lists up all the possible isomorphism classes of indecomposable persistence modules and irreducible morphisms among them, is explicitly derived. In addition, the notion of persistence diagrams is generalized to graphs on the Auslander-Reiten quiver. (iii) An algorithm for computing indecomposable decompositions by using the Auslander- Reiten quiver is presented. (iv) A numerical example to detect robust common topological features is shown.