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

TL;DR

This work tackles the challenge of estimating rare-event probabilities in Gilbert graphs, a class of spatial random networks, by developing a grid-based blocking-region importance-sampling method. It defines a Radon–Nikodym weight that biases point-generation to configurations contributing to the rare event, yielding an estimator with lower variance than naïve or conditional Monte Carlo approaches. The authors prove asymptotic efficiency in two regimes: a fixed sampling window where both CMC and IS achieve bounded relative error, and a growing-window regime where IS attains logarithmic efficiency while CMC does not, supported by rigorous analysis and simulations. Empirical results show dramatic variance reductions for edge-count and degree-related events, validating the method’s practical impact for risk assessment in spatial networks and its potential applicability to more complex network models.

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.

Figures (2)

• Figure 1: Example realizations of Gilbert graphs on a 2-dimensional window $W = [0, \lambda]^2$, where black points represent the nodes, red lines represent the edges, and each circle centered at a node has a unit radius. Small intensity $\kappa$ typically leads to few nodes and few edges as in (a) while large $\kappa$ typically leads to a bigger graph with more edges as in (b).
• Figure 2: Illustration of generating points under the proposed importance sampling method for the edge count problem with $\ell = 10$ on a two-dimensional window. There are $9$ existing points creating $7$ edges. The black region in (a) is the maximum possible blocking region and selecting the next point over the black region in (a) will result in the number edges being more than $10$. Ideally, we would like to generate the next point outside this maximal blocking region (as in (a)). However, identifying that region is difficult. The grid based importance sampling method easily approximates this region from inside as shown in (b).

Theorems & Definitions (25)

• Example 1: Edge Count
• Example 2: Maximum Degree
• Example 3: Maximum Connected Component
• Example 4: Maximum Clique Size
• Example 5: Number of Triangles
• Remark 1: Graphs with a Fixed Number of Nodes
• Proposition 1
• Remark 2: Relationship with Optimal Importance Sampling
• proof : Proof of Proposition \ref{['prop:var_relation']}
• Proposition 2