  We review and extend the recently proposed model of combinatorial quantum
gravity. Contrary to previous discrete approaches, this model is defined on
(regular) random graphs and is driven by a purely combinatorial version of
Ricci curvature, the Ollivier curvature, defined on generic metric spaces
equipped with a Markov chain. It dispenses thus of notions such as simplicial
complexes and Regge calculus and is ideally suited to extend quantum gravity to
combinatorial structures which have a priori nothing to do with geometry.
Indeed, our results show that geometry and general relativity emerge from
random structures in a second-order phase transition due to the condensation of
cycles on random graphs, a critical point that defines quantum gravity
non-perturbatively according to asymptotic safety. In combinatorial quantum
gravity the entropy area law emerges naturally as a consequence of
infinite-dimensional critical behaviour on networks rather than on lattices. We
propose thus that the entropy area law is a signature of the random graph
nature of space-(time) on the smallest scales.
