In this paper, we develop the theory of weighted persistent homology. In 1990, Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then, we study the resulting weighted persistent homology. We show that weighted persistent homology can distinguish between filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can tell when a cycle containing the point is formed or has disappeared.
"Weighted persistent homology." Rocky Mountain J. Math. 48 (8) 2661 - 2687, 2018. https://doi.org/10.1216/RMJ-2018-48-8-2661