Unimodular lattice triangulations as small-world and scale-free random graphs

TL;DR

This paper models lattice triangulations as planar graphs to study their small-world and scale-free properties, revealing insights into real-world network behaviors through Monte Carlo simulations and energy-based ensemble analysis.

Contribution

It introduces a novel approach to analyze lattice triangulations as random graphs exhibiting small-world and scale-free characteristics, using Monte Carlo methods and energy functionals.

Findings

Triangulations have clustering coefficients similar to real networks.

Inverse temperature tuning reveals small-world properties independent of system size.

Quasi-critical tuning indicates scale-free degree distributions for degrees ≥ 5.

Abstract

Real-world networks, e.g. the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte-Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real world graphs, for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees $k \geq 5$. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.