Universality of trap models in the ergodic time scale

Consider a sequence of possibly random graphs GN=(VN,EN), N.1, whose vertices.s have i.i.d. weights {WNx:x.VN} with a distribution belonging to the basin of attraction of an .-stable law, 0<.<1. Let XNt, t.0, be a continuous time simple random walk on GN which waits a mean WNx 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, d.2, random d-regular graphs and the largest component of super-critical Erd.s.Rényi random graphs.