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• W2017687537 abstract "Wireless planar networks have been used to model wireless networks in a tradition that dates back to 1961 to the work of E. N. Gilbert. Indeed the study of connected components in wireless networks was the motivation for his pioneering work that spawned the modern field of continuum percolation theory. Given that node locations in wireless networks are not known, random planar modeling can be used to provide preliminary assessments of important quantities such as range, number of neighbors, power consumption, and connectivity, and issues such as spatial reuse and capacity. In this paper, the problem of connectivity based on nearest neighbors is addressed. The exact threshold function for <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> -coverage is found for wireless networks modeled as <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$n$</tex> points uniformly distributed in a unit square, with every node connecting to its <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$phi _n$</tex> nearest neighbors. A network is called <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> -covered if every node, except those near the boundary, can find one of its <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$phi _n$</tex> nearest neighbors in any sector of angle <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> . For all <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta in (0, 2 pi )$</tex> , if <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$phi _n =(1+delta ) log _ 2pi over 2pi -theta n$</tex> , it is shown that the probability of <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> -coverage goes to one as <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$n$</tex> goes to infinity, for any <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$delta ≫0$</tex> ; on the other hand, if <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$phi _n=(1-delta ) log _ 2pi over 2pi -theta n$</tex> , the probability of <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> -coverage goes to zero. This sharp characterization of <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$theta $</tex> -coverage is used to show, via further geometric arguments, that the network will be connected with probability approaching one if <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>$phi _n=(1+delta ) log _2 n$</tex> . Connections between these results and the performance analysis of wireless networks, especially for routing and topology control algorithms, are discussed. Digital Object Identifier 10.1109/TIT.2006.874384" @default.
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• W2017687537 date "2006-08-01" @default.
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• W2017687537 title "On the<tex>$theta$</tex>-Coverage and Connectivity of Large Random Networks" @default.
• W2017687537 doi "https://doi.org/10.1109/tnet.2006.881854" @default.
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