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 numerical studies, 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.

翻译：定义在欧几里得子空间上的随机几何图（亦称吉尔伯特图）广泛用于模拟各领域中的空间嵌入网络。在这类图中，节点随机分布于欧几里得空间，若任意两节点间的距离小于给定阈值，则存在边连接。精确估计这些图的关键属性（如边数和最大连通分量规模）相关的稀有事件概率，对于评估灾难性事件风险至关重要。然而，这一任务在计算上极具挑战性，尤其针对大规模网络。重要性采样通过将计算资源集中于图的关键区域，提供了可行的解决方案。本文探讨了应用重要性采样方法估计稀有事件概率的可行性，揭示了其在降低方差和提高精度方面的优势。通过渐近分析与数值研究，我们验证了该方法的有效性，为改进吉尔伯特图分析做出贡献，并展示了重要性采样在复杂网络分析中的广泛适用性。