  We study single random walks and the electrical resistance for fractals
obtained as the limit of a sequence of periodic structures. In the long-scale
regime, power laws describe
  both the mean-square displacement of a random walk as a function of time and
the electrical resistance as a function of length. We show that
  the corresponding power-law exponents satisfy the Einstein relation. For
shorter scales, where these exponents depend on length, we find how the
Einstein relation can be generalized to hold locally. All these
  findings were analytically derived and confirmed by numerical
  simulations.
