  Despite the numerous ways now available to quantify which parts or subsystems
of a network are most important, there remains a lack of centrality measures
that are related to the complexity of information flows and are derived
directly from entropy measures. Here, we introduce a ranking of edges based on
how each edge's removal would change a system's von Neumann entropy (VNE),
which is a spectral-entropy measure that has been adapted from quantum
information theory to quantify the complexity of information dynamics over
networks. We show that a direct calculation of such rankings is computationally
inefficient (or unfeasible) for large networks: e.g.\ the scaling is
$\mathcal{O}(N^3)$ per edge for networks with $N$ nodes. To overcome this
limitation, we employ spectral perturbation theory to estimate VNE
perturbations and derive an approximate edge-ranking algorithm that is accurate
and fast to compute, scaling as $\mathcal{O}(N)$ per edge. Focusing on a form
of VNE that is associated with a transport operator $e^{-\beta{ L}}$, where ${
L}$ is a graph Laplacian matrix and $\beta>0$ is a diffusion timescale
parameter, we apply this approach to diverse applications including a network
encoding polarized voting patterns of the 117th U.S. Senate, a multimodal
transportation system including roads and metro lines in London, and a
multiplex brain network encoding correlated human brain activity. Our
experiments highlight situations where the edges that are considered to be most
important for information diffusion complexity can dramatically change as one
considers short, intermediate and long timescales $\beta$ for diffusion.
