 We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.
<meaning-changed>  We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.
 We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs.