  In this paper, we study further properties and applications of weighted
homology and persistent homology. We introduce the Mayer-Vietoris sequence and
generalized Bockstein spectral sequence for weighted homology. For
applications, we show an algorithm to construct a filtration of weighted
simplicial complexes from a weighted network. We also prove a theorem that
allows us to calculate the mod $p^2$ weighted persistent homology given some
information on the mod $p$ weighted persistent homology.
