  We present a simple mathematical framework for the description of the
dynamics of glassy systems in terms of a random walk in a complex energy
landscape pictured as a network of minima. We show how to use the tools
developed for the study of dynamical processes on complex networks, in order to
go beyond mean-field models that consider that all minima are connected to each
other. We consider several possibilities for the transition rates between
minima, and show that in all cases the existence of a glassy phase depends on a
delicate interplay between the network's topology and the relationship between
energy and degree of a minimum. Interestingly, the network's degree
correlations and the details of the transition rates do not play any role in
the existence (nor in the value) of the transition temperature, but have an
impact only on more involved properties. For Glauber or Metropolis rates in
particular, we find that the low-temperature phase can be further divided into
two regions with different scaling properties of the average trapping time.
Overall, our results rationalize and link the empirical findings about
correlations between the energy of the minima and their degree, and should
stimulate further investigations on this issue.
