  Beginning with the Pauli-Fierz theory, we construct a model for
multi-graviton theory. Couplings between gravitons belonging to
nearest-neighbor ``theory spaces'' lead to a discrete mass spectrum. Our model
coincides with the Kaluza-Klein theory whose fifth dimension is latticized.
  We evaluate one-loop vacuum energy in models with a circular latticized extra
dimension as well as with compact continuous dimensions. We find that the
vacuum energy can take a positive value, if the dimension of the continuous
space time is $6, 10,...$. Moreover, since the amount of the vacuum energy can
be an arbitrary small value according to the choice of parameters in the model,
our models is useful to explain the small positive dark energy in the present
universe.
