The birth of geometry in exponential random graphs

TL;DR

This paper introduces explicit exponential random graph models that naturally develop geometric structures like polygons and lattices, avoiding collapse issues and potentially contributing to a graph-based theory of random geometry.

Contribution

The authors construct simple exponential random graph models that produce emergent geometric primitives without explicit programming, advancing the understanding of graph-based random geometries.

Findings

Models exhibit constant vertex degree and form large numbers of polygons.

Emergence of geometric primitives like cubes and lattice fragments.

Models avoid collapse phenomena common in naive Hamiltonian approaches.

Abstract

Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex degree and form very large numbers of simple polygons (triangles or squares). The models avoid the collapse phenomena that plague naive graph Hamiltonians based on triangle or square counts. More than that, statistically significant numbers of other geometric primitives (small pieces of regular lattices, cubes) emerge in our ensemble, even though they are not in any way explicitly pre-programmed into the formulation of the graph Hamiltonian, which only depends on properties of paths of length 2. While much of our motivation comes from hopes to construct a graph-based theory of random geometry (Euclidean quantum gravity), our presentation is completely self-contained within the context of exponential random graph theory, and the range of potential applications is considerably more broad.