Networks as the fundamental constituents of the Universe

Abstract Relations, represented by graph edges, are essential for modelling complex systems across various fields, including physics, computer science, and biology. They provide a universal framework for capturing interactions, dependencies, and pathways, with networks encoding these intricate connections. In this context, we review an approach that uses binary relations as the fundamental constituents of the Universe, utilizing them as building blocks for both space and matter. This model, known as combinatorial quantum gravity, is defined by an ultraviolet continuous fixed point of a statistical model on random networks, governed by the combinatorial Ollivier–Ricci curvature, which acts as a network analogue of the Einstein–Hilbert action. The model exhibits two distinct phases separated by this fixed point, a geometric and a random phase, representing space and matter, respectively. At weak coupling and on large scales, the network organizes into a holographic surface whose collective state encodes both an emergent 3D space and the matter distributed in it. The Einstein equations emerge as constitutive relations expressing matter in terms of fundamental network degrees of freedom while dynamics in a comoving frame is governed by relativistic quantum mechanics. Quantum mechanics, however is an effective theory breaking down at the scale of the radius of curvature of the holographic network. On smaller scales, not only relativistic invariance is lost but also the Lorentzian signature of space-time. Finally, the manifold nature of space-time breaks down on the Planck length, where the random character of the fundamental network on the smallest scales becomes apparent. The network model seems to naturally encode several of the large-distance features of cosmology, albeit still at a qualitative level. The holographic property of black holes arises intrinsically from the expander nature of random regular graphs. There is a natural mechanism to resolve the cosmological constant problem and dark matter appears naturally as a metastable allotrope in the network fabric of space-time. In this model, both gravity and quantum mechanics are macroscopic statistical effects reflecting the free energy minimization of fundamental binary degrees of freedom.