Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

Deriving the critical density (which is equivalent to deriving the critical radius or power) to achieve coverage and/or connectivity for random deployments is a fundamental problem in the area of wireless networks. The probabilistic conditions normally derived, however, have limited appeal among practitioners because they are often asymptotic, i.e., they only make high probability guarantees in the limit of large system sizes. Such conditions are not very useful in practice since deployment regions are always finite. Another major limitation of most existing work on coverage and connectivity is their focus on thick deployment regions (such as a square or a disk). There is no existing work (including traditional percolation theory) that derives critical densities for thin strips (or annuli).In this paper, we address both of these shortcomings by introducing new techniques for deriving reliable density estimates for finite regions (including thin strips). We apply our techniques to solve the open problem of deriving reliable density estimates for achieving barrier coverage and connectivity in thin strips, where sensors are deployed as a barrier to detect moving objects and phenomena. We use simulations to show that our estimates are accurate even for small deployment regions. Our techniques bridge the gap between theory and practice in the area of coverage and connectivity, since the results can now be readily used in real-life deployments.

Let 𝓅 be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With Gn, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

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,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

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