Rare event probabilities in Random Geometric Graphs

TL;DR

This work analyzes rare events in high-dimensional spherical and Gaussian random geometric graphs, focusing on when the graph is a complete clique and when the edge count is atypically large. By coupling the spherical and Gaussian models and employing symmetric rearrangement alongside cap-based geometric arguments, the authors derive sharp large-deviation rates that depend on $n$ and $d$, including regime-specific bounds and matching constants in several high-dimension regimes. The results advance understanding of how latent geometry influences extremal graph properties and offer tools for assessing robustness and anomaly detection in geometric networks. Overall, the paper provides a rigorous, dimension-aware large-deviation framework for two canonical rare-events in dense, high-dimensional RGGs with edge probability $p$.

Abstract

In this paper, we study rare events in spherical and Gaussian random geometric graphs in high dimensions. In these models, the vertices correspond to points sampled uniformly at random on the $d$ dimensional unit sphere or correspond to $d$ dimensional standard Gaussian vectors, and edges are added between two vertices if the inner-product between their corresponding points are greater than a threshold $t_p$, chosen such that the probability of having an edge is equal to $p$. We focus on two problems: (a) the probability that the RGG is a complete graph, and (b) the probability of observing an atypically large number of edges. We obtain asymptotically exponential decay rates depending on $n$ and $d$ of the probabilities of these rare events through a combination of geometric and probabilistic arguments.

Theorems & Definitions (56)

• Remark 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.1
• Lemma 1
• proof
• Lemma 2
• proof