Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Abstract

We consider the Voronoi tessellation induced by a homogeneous and stationary Poisson point process of intensity λ > 0 in a quadrant, where the two half-axes represent boundaries. We show that the mean cell size is less than λ −1 when the seed is located exactly at the boundary, and it can be larger than λ −1 when the seed lies close to the boundary. In addition, we calculate the second moment of the cell size at two locations: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. In both cases, we illustrate that the two-parameter Gamma distribution, with location-dependent parameters, provides a good fit. As a potential application, we use the Gamma approximations to study the degree distribution for secure in-connectivity in wireless sensor networks deployed over a bounded domain. Index TermsPhysical layer security, Poisson Voronoi tessellations, stochastic geometry I. INTRODUCTIONA random tessellation is a random subdivision of a space into disjoint regions or cells C i , see [1], [2] for a formal definition. Perhaps the most basic random tessellation model partitions the plane R 2 into Voronoi cells. In order to construct them, a set of random nuclei (or seeds) S i are first distributed, and then, the locations of the plane are associated with the nearest seed for the Euclidean distance. The boundaries of the Voronoi cells are equidistant to the two nearest K. Koufos and C.P. Dettmann are with the

Statistical properties of Poisson-Voronoi tessellation cells in bounded regions

“…Ripley [2] discusses the importance of space subdivision methods to investigate spatial splines, and gives examples of different spatial point patterns for both simulated and real data to relate the subject to the estimation of distributions of the locations within a region using the Voronoi tessellation. The Voronoi tessellation has been applied in different sciences such as in seismology where Schoenberg [3] investigated the distribution of cell areas in a Voronoi tessellation based on the locations of earthquakes in Southern California; astronomy [4][5][6] to discover how galaxies are distributed in space; to investigate the conditions of the habitat of animals when they are establishing territories [7]; in agriculture for maximal weed suppression in plant crops [8] and to study atomic crystals [9], liquids [10], glasses [11], and wireless networks [12,13]. An application of constrained Voronoi tessellation is used in micro-structure modeling [14] where a new space subdivision method is introduced using inverse Monte Carlo based on conditions such as moving the randomly placed points until their geometric features obey a particular distribution.…”

Finding wombling boundaries in LHC data with Voronoi and Delaunay tessellations

“…For concreteness, we shall choose our field of view to be the unit square, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, although data will be generated beyond the boundaries of the unit squarethis will eliminate any spurious boundary effects like clipping which would modify the statistical properties of the Voronoi cells near the boundaries [64]. Following refs.…”

Boundaries as an Enhancement Technique for Physical Layer Security

“…The information theoretic approaches [9]- [11] and the analysis using stochastic geometry [12]- [20] assume that the locations of the transmitters and the eavesdroppers follow the uniform distribution in the infinite plane. To the best of our knowledge, the only available studies considering the impact of boundaries on secrecy performance are [26], [27]. The study in [26] neglects the interference effects, and shows that the mean in-and out-connectivity degrees with PLS in a quadrant are not necessarily equal, unlike in the infinite plane.…”

“…To the best of our knowledge, the only available studies considering the impact of boundaries on secrecy performance are [26], [27]. The study in [26] neglects the interference effects, and shows that the mean in-and out-connectivity degrees with PLS in a quadrant are not necessarily equal, unlike in the infinite plane. The study in [27] considers a transmitter-receiver pair and a Poisson Point Process (PPP) for the locations of eavesdroppers inside an L-sided convex polygon.…”