Given a graph G with n nodes and adjacency matrix A, the Katz Centrality measure, K(G), is given by $K(G)[i] = \sum_{k=1}^{\infty}\sum_{j=1}^{n}\alpha^k(A^k)_{ji}$ s.t. $\alpha$ is an attenuating factor and $\alpha \in (0,1)$ i.e. it measures the degree of the node plus the sum over k of the number of paths of length k with diminishing value.
Question: is there an adaptation for weighted graphs? (Or, is there is an adjacency matrix multiplication that can be used to find the sum of all paths of length k starting at a given node in a graph?)