In the ordinary theory of Sobolev spaces on domains of \(\mathbb{R}^{n}\), the \(p\)-energy is defined as the integral of \(\vert\nabla f\vert^{p}\). In this book, the author tries to construct a \(p\)-energy on compact metric spaces as a scaling limit of discrete \(p\)-energies on a series of graphs approximating the original space. In conclusion, the author proposes 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 he constructs a local regular Dirichlet form and shows 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, the author presents new classes of square-based, self-similar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.