  To monitor risk in temporal financial networks, we need to understand how
individual behaviours affect the global evolution of networks. Here we define a
structural importance metric - which we denote as $l_e$ - for the edges of a
network. The metric is based on perturbing the adjacency matrix and observing
the resultant change in its largest eigenvalues. We then propose a model of
network evolution where this metric controls the probabilities of subsequent
edge changes. We show using synthetic data how the parameters of the model are
related to the capability of predicting whether an edge will change from its
value of $l_e$. We then estimate the model parameters associated with five real
financial and social networks, and we study their predictability. These methods
have application in financial regulation whereby it is important to understand
how individual changes to financial networks will impact their global
behaviour. It also provides fundamental insights into spectral predictability
in networks, and it demonstrates how spectral perturbations can be a useful
tool in understanding the interplay between micro and macro features of
networks.
