Full Connectivity: Corners, Edges and Faces

Coon, Justin P.; Dettmann, Carl P.; Georgiou, Orestis

doi:10.1007/s10955-012-0493-y

Cited by 58 publications

(140 citation statements)

References 44 publications

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“…That is, in d = 2 the larger number of nodes in the bulk dominates the lower probability of links for nodes near the boundary. However, the present authors [16] have pointed out that for practical purposes, namely approximating P f c in a realistic system, the size is not exponentially large, and the bulk, edges, or corners may dominate the connection probability [16] depending on the density. Thus, we are interested in results involving more general limiting processes, as well as useful approximations for finite cases.…”

Section: A Backgroundmentioning

confidence: 74%

“…The connection (and hence k-connection) probability P f c can then be written in "semigeneral" form [16] as a sum of contributions from different boundary elements, (9) below. To illustrate the notation, we give the case of a square domain: …”

Section: B Summary Of New Resultsmentioning

confidence: 99%

“…The earlier paper [16] gave special cases of these, namely G π/2 3,i and G ω 2,i , with a typo for G ω 2,1 . The other paper with directly comparable results is Ref.…”

Section: Comparison With Previous Results and Numericsmentioning

confidence: 99%

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Random geometric graphs with general connection functions

Dettmann

Georgiou

2016

Phys. Rev. E

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In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H (r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected; that is, every node is linked to every other node in a multihop fashion. Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.

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Section: A Backgroundmentioning

confidence: 74%

Section: B Summary Of New Resultsmentioning

confidence: 99%

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Random geometric graphs with general connection functions

Dettmann

Georgiou

2016

Phys. Rev. E

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“…We model a wireless network as a random geometric graph (RGG) with a probabilistic pair connection rule [25], [26]. Consider a set V n = {1, .…”

Section: Model and Assumptionsmentioning

confidence: 99%

“…where, in this context, r 0 signifies the typical connection range (rather than the maximum), which encompasses physical system characteristics, such as the transmit power, wavelength, and the noise figure [22], [23], [25]. The parameter η is the path loss exponent in this model, and thus it typically takes on values in the range 2 ≤ η ≤ 5; mathematically, it simply controls the stretch of the decaying exponential, and by letting η → ∞, we recover the hard connection model.…”

Section: B Graph Entropy Conditioned On Node Positionsmentioning

confidence: 99%

On the conditional entropy of wireless networks

Coon

Badiu

Gündüz

2018

2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)

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Abstract-The characterization of topological uncertainty in wireless networks using the formalism of graph entropy has received interest in the spatial networks community. In this paper, we develop lower bounds on the entropy of a wireless network by conditioning on potential network observables. Two approaches are considered: 1) conditioning on subgraphs, and 2) conditioning on node positions. The first approach is shown to yield a relatively tight bound on the network entropy. The second yields a loose bound, in general, but it provides insight into the dependence between node positions (modelled using a homogenous binomial point process in this work) and the network topology.

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Distance distribution between two random points in arbitrary polygons

Pure

Durrani

Tong

et al. 2021

Math Methods in App Sciences

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Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically.

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Full Connectivity: Corners, Edges and Faces

Cited by 58 publications

References 44 publications

Random geometric graphs with general connection functions

Random geometric graphs with general connection functions

On the conditional entropy of wireless networks

Distance distribution between two random points in arbitrary polygons

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