# Symmetric motifs in random geometric graphs

## TL;DR

This paper investigates symmetric motifs in random geometric graphs, analyzing their occurrence probabilities, spectral signatures, and the minimum separation distances across different limits and dimensions.

## Contribution

It provides a detailed probabilistic and spectral analysis of symmetric motifs, including their prevalence and the minimum separation distance in various limits.

## Key findings

- Symmetric motifs are prevalent and produce distinct spectral peaks.
- Probability of closest nodes being symmetric approaches one in the thermodynamic limit.
- In the intensive limit, symmetry probability depends on the dimension.

## Abstract

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen-Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit the probability that the closest nodes are symmetric approaches one, whilst in the intensive limit this probability depends upon the dimension.