Networks as the fundamental constituents of the universe

TL;DR

The work proposes a gravity-and-quantum framework grounded in discrete network geometry, where binary relations and curvature measures on graphs underpin emergent space, matter, and dynamics. By employing Ollivier-Ricci curvature as a discrete Einstein-Hilbert analogue and analyzing a cycle-condensation transition, it demonstrates how a geometric, holographic 2D surface can emerge with a (1+3)D bulk supporting relativistic quantum mechanics at large scales. Einstein equations arise as constitutive relations from the network's free-energy minimization, while quantum behavior, holographic black holes, and dark matter emerge as macroscopic statistical effects of the underlying graph dynamics. The model provides a qualitative account of large-scale cosmology, black-hole holography, and dark matter as allotropes of the network fabric, offering a discrete, background-independent route toward quantum gravity with testable geometric features.

Abstract

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. The model 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.

Figures (9)

• Figure 1: Excluded subgraph characterisation of the hard-core condition. Listed pairs for each subgraph give the types of short cycles sharing more than one edge
• Figure 2: For $N=100$ vertices we generate random Erdös-Rényi graphs, varying the link probability $p$. For a collection of $10$ graphs for each value of $p$, we compute the average fractional difference between $\langle k\rangle \kappa^{OR}_{ij}$ and $\kappa^{FR}_{ij}$, which we plot against the average of $I_{ij}=\langle k\rangle-2-\triangle_{ij}-\square_{ij} -\pentagon_{ij}$. The blue dots represent the raw simulation values, with the blue line representing the linear regression fit, shading to cover the 95% confidence interval for the regression.
• Figure 3: The order parameter $S/N$ for the random-to-geometric phase transition for $D=2$.
• Figure 4: A two-torus at a small but finite value of the coupling $g$ in the geometric phase
• Figure 5: The typical behaviour of the connected correlation function between the numbers of squares at graph distance $d$ in the geometric phase.

Theorems & Definitions (2)

• Definition 1
• Definition 2