  We introduce group field theory networks as a generalization of spin networks
and of (symmetric) random tensor networks and provide a statistical computation
of the R\'enyi entropy for a bipartite network state using the partition
function of a simple interacting group field theory. The expectation value of
the entanglement entropy is calculated by an expansion into stranded Feynman
graphs and is shown to be captured by a Ryu- Takayanagi formula. For a simple
interacting group field theory, we can prove the linear corrections, given by a
polynomial perturbation of the Gaussian measure, to be negligible for a broad
class of networks.
