Connectivity in a random interval graph with access points

Shang, Yilun

doi:10.1016/j.ipl.2009.01.002

Cited by 12 publications

(7 citation statements)

References 9 publications

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“…In [10] the connectivity of G(n, z, r) with heterogeneous r is analyzed. A closed-form connectivity probability of G(n, z, r) with access points is worked out in [11] by virtue of some combinatorial identities. Ref.…”

Section: Definitionmentioning

confidence: 99%

A note on the 2-connectivity in one-dimensional ad hoc networks

Shang

2010

Sci. China Inf. Sci.

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2-connectivity is the basic graph theoretical metric for fault tolerance to node failures, and it is an extremely desirable property in network design. For a one-dimensional finite ad hoc network formed by n nodes uniformly and independently distributed in a closed interval [0, z] (z ∈ R + ), we give a closed form expression for the probability that a connected one-dimensional finite ad hoc network becomes disconnected after removing a node. We also sketch a procedure of finding the exact formula for 2-connectivity in a one-dimensional ad hoc network.

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Section: Definitionmentioning

confidence: 99%

A note on the 2-connectivity in one-dimensional ad hoc networks

Shang

2010

Sci. China Inf. Sci.

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“…Firstly, as random spatial models in the physics of complex systems, see for example the 1d soft random geometric graph [10] used in complex networks by Krioukov et al in network geometry [21]. Secondly, in Poisson-Boolean continuum percolation [13,[22][23][24], and thirdly, in vehicular communications [10,11,14,[25][26][27][28][29][30][31][32][32][33][34][35][36][37][38][39]. For 1d spatial models similar to the 1d RGG, see the 1d exponential random geometric graph [40], random interval graphs where the connection is between overlapping intervals of random length [28,41], or various models of one-dimensional mathematical physics [13,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…Secondly, in Poisson-Boolean continuum percolation [13,[22][23][24], and thirdly, in vehicular communications [10,11,14,[25][26][27][28][29][30][31][32][32][33][34][35][36][37][38][39]. For 1d spatial models similar to the 1d RGG, see the 1d exponential random geometric graph [40], random interval graphs where the connection is between overlapping intervals of random length [28,41], or various models of one-dimensional mathematical physics [13,17,18]. For a historical introduction to the similar problem of covering a line by random overlapping intervals, see Domb [42].…”

Section: Introductionmentioning

confidence: 99%

Connectivity of 1d random geometric graphs

Kartun-Giles¹,

Koufos²,

Privault³

2021

Preprint

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A 1d random geometric graph (1d RGG) is built by joining a random sample of n points from an interval of the real line with probability p. We count the number of k-hop paths between two vertices of the graph in the case where the space is the 1d interval [0, 1]. We show how the k-hop path count between two vertices at Euclidean distance |x − y| is in bijection with the volume enclosed by a uniformly random d-dimensional lattice path joining the corners of a (k − 1)-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of this k-hop path count as a sum over lattice paths, incorporating the idea of restricted integer partitions with limited number of parts. We therefore demonstrate and describe an important link between spatial random graphs, and lattice path combinatorics, where the d-dimensional lattice paths correspond to spatial permutations of the geometric points on the line.

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“…Lin introduced circular trapezoid graphs (CTG), which constitute a proper superclass of trapezoid graphs and circular-arc graphs [10]. He also presented ( ) 2 log log O n n time and ( ) 2 log O n n time algorithms for the maximum weighted independent set and the minimum weighted independent dominating set on CTGs, respectively [10]. In this paper, we designed an…”

Section: Introductionmentioning

confidence: 99%