  We show that, in discrete models of quantum gravity, emergent geometric space
can be viewed as the entanglement pattern in a mixed quantum state of the
"universe", characterized by a universal topological network entanglement. As a
concrete example we analyze the recently proposed model in which geometry
emerges due to the condensation of 4-cycles in random regular bipartite graphs,
driven by the combinatorial Ollivier-Ricci curvature. Using this model we show
that the emergence of geometric order decreases the entanglement entropy of
random configurations. The lowest geometric entanglement entropy is realized in
four dimensions.
