A critical constant for the κ-nearest-neighbour model
Let P be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊clogn⌋ is disconnected with probability tending to 1 as n → 8 and, for c > ccrit, Gn,⌊clogn⌋ is connected with probability tending to 1 as n → 8. This answers a question posed in Balister et al. (2005). © Applied Probability Trust 2009.
Advances in Applied Probability
Balister, P., Bollobás, B., Sarkar, A., & Walters, M. (2009). A critical constant for the κ-nearest-neighbour model. Advances in Applied Probability, 41 (1), 1-12. https://doi.org/10.1239/aap/1240319574