  We introduce a new invariant defined on the vertices of a given filtered
simplicial complex, called codensity, which controls the impact of removing
vertices on persistent homology. We achieve this control through the use of an
interleaving type of distance between fitered simplicial complexes. We study
the special case of Vietoris-Rips filtrations and show that our bounds offer a
significant improvement over the immediate bounds coming from considerations
related to the Gromov-Hausdorff distance. Based on these ideas we give an
iterative method for the practical simplification of filtered simplicial
complexes.
  As a byproduct of our analysis we identify a notion of core of a filtered
simplicial complex which admits the interpretation as a minimalistic simplicial
filtration which retains all the persistent homology information.
