  The slow relaxation and aging of glassy systems can be modelled as a Markov
process on a simplified rough energy landscape: energy minima where the system
tends to get trapped are taken as nodes of a random network, and the dynamics
are governed by the transition rates among these. In this work we consider the
case of purely activated dynamics, where the transition rates only depend on
the depth of the departing trap. The random connectivity and the disorder in
the trap depths make it impossible to solve the model analytically, so we base
our analysis on the spectrum of eigenvalues $\lambda$ of the master operator.
We compute the local density of states $\rho(\lambda|\tau)$ for traps with a
fixed lifetime $\tau$ by means of the cavity method. This exhibits a power law
behaviour $\rho(\lambda|\tau)\sim\tau|\lambda|^T$ in the regime of small
relaxation rates $|\lambda|$, which we rationalize using a simple analytical
approximation. In the time domain, we find that the probabilities of return to
a starting node have a power law-tail that is determined by the distribution of
excursion times $F(t)\sim t^{-(T+1)}$. We show that these results arise only by
the combination of finite configuration space connectivity and glassy disorder,
and interpret them in a simple physical picture dominated by jumps to deep
neighbouring traps.
