Moments of <i>k</i>-hop counts in the random-connection model

Privault, Nicolas

doi:10.1017/jpr.2019.63

Cited by 9 publications

(9 citation statements)

References 18 publications

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“…However, the distribution of the number of nodes with a two-hop path is not treated therein, neither asymptotic equivalents for ultra-dense and sparse networks are derived, as we will do in this paper. Finally, in [26], [27] the moments of the number of k-hop paths to the origin in a 2D random connection model are computed, which is a different metric as compared to that studied in the present paper, where the moments of the number of nodes in the graph with a k-hop path to the origin are calculated.…”

Section: Related Workmentioning

confidence: 99%

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

Kartun-Giles¹,

Koufos²,

Xiao³

et al. 2020

Preprint

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We compute the expected number of cars that have at least one two-hop path to a fixed roadside unit in a one-dimensional vehicular ad hoc network in which other cars can be used as relays to reach a roadside unit when they do not have a reliable direct link. The pairwise channels between cars experience Rayleigh fading in the random connection model, and so exist, with probability function of the mutual distance between the cars, or between the cars and the roadside unit. We derive exact equivalents for this expected number of cars when the car density ρ tends to zero and to infinity, and determine its behaviour using an infinite oscillating power series in ρ, which is accurate for all regimes.We also corroborate those findings to a realistic situation, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the origin. The probability mass function appears to be well fitted by a Gaussian approximation with mean equal to the expected number of cars with two hops to the origin.

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Section: Related Workmentioning

confidence: 99%

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

Kartun-Giles¹,

Koufos²,

Xiao³

et al. 2020

Preprint

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“…The moments of k-hop counts in the random-connection model have been expressed in [Pri19] as summations over non-flat partition diagrams, however, those expressions are difficult to apply to the derivation of explicit bounds. In this paper we use a different approach based on the representation of k-hop counts in terms of multiple Poisson stochastic integrals, which allows us to derive explicit expressions for moments and cumulants of all orders by recursive formulas.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we specialize those expressions when the k-hops are made of a single node per cell. Section 4 presents moment expressions in terms of sums over non-flat partitions based on results of [Pri19]. Sections 5 and 6 develop recursive expressions for the explicit calculation of joint moments and cumulants of any order.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model

Privault¹

2022

Preprint

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We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein method we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants are provided in appendix.

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“…The soft case [11,14,[29][30][31][32] remains unsolved. The value of understanding the mathematics of the 1d case is well motivated based on some recent problems in spatial complex networks [2,6,14,[44][45][46][47][48], particularly betweenness centrality [44], which involves counting the number of paths between two nodes in a complex network [2,45]. Therefore, in this article, we focus on counting the integer number σ k of k-hop paths which run between two fixed vertices of the 1d RGG.…”

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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Moments of k-hop counts in the random-connection model

Cited by 9 publications

References 18 publications

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model

Connectivity of 1d random geometric graphs

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