  In this paper we propose a method to compute the solution to the fractional
diffusion equation on directed networks, which can be expressed in terms of the
graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a
non-analytic function involving fractional powers and $\boldsymbol{b}$ is a
given vector. The graph Laplacian is a singular matrix, causing Krylov methods
for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this
difficulty and achieve faster convergence, we use rational Krylov methods
applied to a desingularized version of the graph Laplacian, obtained with
either a rank-one shift or a projection on a subspace.
