  This article is in the spirit of our work on the consequences of the Regge
calculus, where some edge length scale arises as an optimal initial point of
the perturbative expansion after functional integration over connection.
  Now consider the perturbative expansion itself. To obtain an algorithmizable
diagram technique, we consider the simplest periodic simplicial structure with
a frozen part of the variables ("hypercubic"). After functional integration
over connection, the system is described by the metric $g_{\lambda \mu}$ at the
sites.
  We parameterize $g_{\lambda \mu}$ so that the functional measure becomes
Lebesgue. The discrete diagrams are free from ultraviolet divergences and
reproduce (for ordinary, non-Planck external momenta) those continuum
counterparts that are finite. We give the parametrization of $g_{\lambda \mu}$
up to terms, providing, in particular, additional three-graviton and
two-graviton-two-matter vertices, which can give additional one-loop
corrections to the Newtonian potential.
  The edge length scale is $\sim \sqrt{ \eta }$, where $\eta$ defines the free
factor $ ( - \det \| g_{\lambda \mu} \| )^{ \eta / 2}$ in the measure and
should be a large parameter to ensure the true action after integration over
connection. We verify the important fact that the perturbative expansion does
not contain increasing powers of $\eta$ if its initial point is chosen close
enough to the maximum point of the measure, thus justifying this choice.
  Discrete propagators depend on the Barbero-Immirzi parameter $\gamma$, which
determines the ratio of timelike and spacelike elementary length scales. The
existing estimates of $\gamma$ allow the propagator poles to have real energy
for any (real) spatial momenta.
