The decay centrality of a vertex $v$ in a graph $G$ with respect to a parameter $\delta \in (0,1)$ is a polynomial in δ such that for fixed $k$ the coefficient of $\delta ^k$ is equal to the number of vertices of $G$ at distance $k$ from $v$. This invariant (introduced independently by Jackson and Wolinsky in 1996 and Dangalchev in 2011) is considered as a replacement for the closeness centrality for graphs, however its unstability was pointed out by Yang and Zhuhadar in 2011. We explore mathematical properties of decay centrality depending on the choice of parameter δ and the stability of vertex ranking based on this centrality index.