  For many functions of matrices $f(A)$, it is known that their entries exhibit
a rapid -- often exponential or even superexponential -- decay away from the
sparsity pattern of the matrix $A$. In this paper we specifically focus on the
class of Bernstein functions, which contains the fractional powers $A^\alpha$,
$\alpha \in (0,1)$ as an important special case, and derive new decay bounds by
exploiting known results for the matrix exponential in conjunction with the
L\'evy--Khintchine integral representation. As a particular special case, we
find a result concerning the power law decay of the strength of connection in
nonlocal network dynamics described by the fractional graph Laplacian, which
improves upon known results from the literature by doubling the exponent in the
power law.
