  The leading eigenvalue $\lambda$ of the adjacency matrix of a graph exerts
much influence on the behavior of dynamical processes on that graph. It is thus
relevant to relate notions of the importance (specifically, centrality
measures) of network structures to $\lambda$ and its associated eigenvector. We
study a previously derived measure of edge importance known as "dynamical
importance", which estimates how much $\lambda$ changes when one removes an
edge from a graph or adds an edge to it. We examine the accuracy of this
estimate for different network structures and compare it to the true change in
$\lambda$ after an edge removal or edge addition. We then derive a first-order
approximation of the change in the leading eigenvector. We also consider the
effects of edge additions on Kuramoto dynamics on networks, and we express the
Kuramoto order parameter in terms of dynamical importance. Through our analysis
and computational experiments, we find that studying dynamical importance can
improve understanding of the relationship between network perturbations and
dynamical processes on networks.
