Rare Events in Random Geometric Graphs

TL;DR

This paper develops and compares advanced Monte Carlo and importance sampling methods for accurately estimating rare-event probabilities related to the number of edges in random geometric graphs, with proven variance reductions.

Contribution

It introduces new conditional Monte Carlo algorithms and importance sampling techniques based on large deviations theory for better rare-event probability estimation in geometric graphs.

Findings

Conditional Monte Carlo reduces variance compared to crude Monte Carlo.

Importance sampling with Gibbsian processes further decreases estimation variance.

Methods are validated through simulation studies demonstrating significant improvements.

Abstract

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we leverage conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.