Rare Events in Random Geometric Graphs

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.

[1]  P. Billingsley,et al.  Probability and Measure , 1980 .

[2]  M. Lomonosov,et al.  Combinatorics of reliability Monte Carlo , 1999 .

[3]  J. Yukich,et al.  Large deviation principles for Euclidean functionals and other nearly additive processes , 2001 .

[4]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[5]  Dirk P. Kroese,et al.  Improved algorithms for rare event simulation with heavy tails , 2006, Advances in Applied Probability.

[6]  P. Glynn,et al.  Efficient rare-event simulation for the maximum of heavy-tailed random walks , 2008, 0808.2731.

[7]  François Baccelli,et al.  Stochastic Geometry and Wireless Networks, Volume 1: Theory , 2009, Found. Trends Netw..

[8]  Dirk P. Kroese,et al.  Handbook of Monte Carlo Methods , 2011 .

[9]  A. Baddeley,et al.  Approximating the moments of a spatial point process , 2012 .

[10]  L. Rojas-Nandayapa.  A review of conditional rare event simulation for tail probabilities of heavy tailed random variables , 2013 .

[11]  Shankar Bhamidi,et al.  The importance sampling technique for understanding rare events in Erd\H{o}s-R\'enyi random graphs , 2013, 1302.6551.

[12]  Volker Schmidt,et al.  A General Framework for Consistent Estimation of Charge Transport Properties via Random Walks in Random Environments , 2014, Multiscale Model. Simul..

[13]  S. Chatterjee,et al.  Localization in random geometric graphs with too many edges , 2014, The Annals of Probability.

[14]  Dirk P. Kroese,et al.  Improved Sampling Plans for Combinatorial Invariants of Coherent Systems , 2016, IEEE Transactions on Reliability.

[15]  Mathew D. Penrose,et al.  Lectures on the Poisson Process , 2017 .

[16]  Holger Paul Keeler,et al.  Disruptive events in high-density cellular networks , 2018, 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt).