  Trap models describe glassy dynamics as a stochastic process on a network of
configurations representing local energy minima. We study within this class the
paradigmatic Barrat-M\'ezard model, which has Glauber transition rates. Our
focus is on the effects of the network connectivity, where we go beyond the
usual mean field (fully connected) approximation and consider sparse networks,
specifically random regular graphs. We obtain the spectral density of
relaxation rates of the master operator using the cavity method, revealing very
rich behaviour as a function of network connectivity $c$ and temperature $T$.
We trace this back to a crossover from initially entropic barriers, resulting
from a paucity of downhill directions, to energy barriers that govern the
escape from local minima at long times. The insights gained are used to
rationalize the relaxation of the energy after a quench from high $T$, as well
as the corresponding correlation and persistence functions.
