We introduce weighted versions of the classical Čech and Vietoris–Rips complexes. We show that a version of the Vietoris–Rips lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.
"Weighted persistent homology." Involve 12 (5) 823 - 837, 2019. https://doi.org/10.2140/involve.2019.12.823