Consider a sequence of possibly random graphs GNβ=(VNβ,ENβ), Nβ₯1,
whose vertices's have i.i.d. weights {WxNβ:xβVNβ} with a distribution
belonging to the basin of attraction of an Ξ±-stable law, 0<Ξ±<1.
Let XtNβ, tβ₯0, be a continuous time simple random walk on GNβ which
waits a \emph{mean} WxNβ exponential time at each vertex x. Under
considerably general hypotheses, we prove that in the ergodic time scale this
trap model converges in an appropriate topology to a K-process. We apply this
result to a class of graphs which includes the hypercube, the d-dimensional
torus, dβ₯2, random d-regular graphs and the largest component of
super-critical Erd\"os-R\'enyi random graphs