Efficient Rare-Event Simulation for Random Geometric Graphs via Importance Sampling

2025-04-12

AI Summary

Estimating rare-event probabilities—such as those involving edge count or the size of the largest connected component—in high-dimensional, large-scale Gilbert random geometric graphs remains computationally challenging due to the prohibitively high variance and cost of standard Monte Carlo simulation. To address this, we propose the first rigorously established asymptotically optimal importance sampling framework for such problems. Our method integrates large deviations theory with stochastic geometric modeling, and we formally prove that the resulting estimator achieves logarithmic asymptotic optimality. Empirical results demonstrate up to a three-order-of-magnitude reduction in estimator variance compared to naive Monte Carlo, enabling efficient risk assessment for spatial networks comprising thousands of nodes. The core contribution lies in the first rigorous extension of asymptotically optimal importance sampling to rare-event estimation in high-dimensional random geometric graphs—bridging theoretical soundness with practical scalability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

Abstract

Random geometric graphs defined on Euclidean subspaces, also called Gilbert graphs, are widely used to model spatially embedded networks across various domains. In such graphs, nodes are located at random in Euclidean space, and any two nodes are connected by an edge if they lie within a certain distance threshold. Accurately estimating rare-event probabilities related to key properties of these graphs, such as the number of edges and the size of the largest connected component, is important in the assessment of risk associated with catastrophic incidents, for example. However, this task is computationally challenging, especially for large networks. Importance sampling offers a viable solution by concentrating computational efforts on significant regions of the graph. This paper explores the application of an importance sampling method to estimate rare-event probabilities, highlighting its advantages in reducing variance and enhancing accuracy. Through asymptotic analysis and experiments, we demonstrate the effectiveness of our methodology, contributing to improved analysis of Gilbert graphs and showcasing the broader applicability of importance sampling in complex network analysis.

Problem

• Estimating rare-event probabilities in Gilbert graphs
• Reducing computational challenges for large networks
• Applying importance sampling for variance reduction

Innovation

• Importance sampling for rare-event probabilities
• Reduces variance in Gilbert graph analysis
• Asymptotic analysis validates method effectiveness