Universality of trap models in the ergodic time scale

Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ 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, $d\ge 2$, random $d$-regular graphs and the largest component of super-critical Erd\"os-R\'enyi random graphs.