  In the ordinary theory of Sobolev spaces on domains of $R^n$, the $p$-energy
is defined as the integral of $|\nabla{f}|^p$. In this paper, we try to
construct $p$-energy on compact metric spaces as a scaling limit of discrete
$p$-energies on a series of graphs approximating the original space. In
conclusion, we propose a notion called conductive homogeneity under which one
can construct a reasonable $p$-energy if $p$ is greater than the Ahlfors
regular conformal dimension of the space. In particular, if $p = 2$, then we
construct a local regular Dirichlet form and show that the heat kernel
associated with the Dirichlet form satisfies upper and lower sub-Gaussian type
heat kernel estimates. As examples of conductively homogeneous spaces, we
present a new class of square-based self-similar sets and rationally ramified
Sierpinski cross, where no diffusion was constructed before.
