  We consider the task of updating a matrix function $f(A)$ when the matrix
$A\in{\mathbb C}^{n \times n}$ is subject to a low-rank modification. In other
words, we aim at approximating $f(A+D)-f(A)$ for a matrix $D$ of rank $k \ll
n$. The approach proposed in this paper attains efficiency by projecting onto
tensorized Krylov subspaces produced by matrix-vector multiplications with $A$
and $A^*$. We prove the approximations obtained from $m$ steps of the proposed
methods are exact if $f$ is a polynomial of degree at most $m$ and use this as
a basis for proving a variety of convergence results, in particular for the
matrix exponential and for Markov functions. We illustrate the performance of
our method by considering various examples from network analysis, where our
approach can be used to cheaply update centrality and communicability measures.
