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

Topological Network Entanglement as Order Parameter for the Emergence of Geometry