  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.
