Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Yao, Chang-Long; Chen, Ge; Guo, Tong

doi:10.1017/s0021900200007695

Cited by 12 publications

(18 citation statements)

References 12 publications

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“…is the Palm probability, that a randomly selected user is connected to infinity in the Gilbert graph. Finally, in view of sub-additivity of the number of hops and the corresponding results on the Poisson Boolean model (PBM), see [13], it is reasonable to assume the existence of a deterministic stretch factor µ(λ, r, γ) > 1/r given by…”

Section: Network Modelmentioning

confidence: 99%

Percolation for D2D networks on street systems

Cali

Gafur

Hirsch

et al. 2018

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

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We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.

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Section: Network Modelmentioning

confidence: 99%

Percolation for D2D networks on street systems

Cali

Gafur

Hirsch

et al. 2018

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

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“…Loosely speaking, for distant points in the infinite connected component, the chemical distance is approximately proportional to the Euclidean distance, where the proportionality factor is called the time constant. The extension of this result to the setting of continuum percolation [15] will be the major tool for establishing the distributional limit of the rescaled minimum number of hops needed to connect a user to a base station.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…, 0) is the first standard unit vector in R d . Using Kingman's subadditive ergodic theorem, it was shown in [15] that there exists a real number μ ∈ (0, ∞) such that almost surely, lim n→∞ n −1 D n = μ; see also [2] for the corresponding statement on the lattice.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bounded-hop percolation and wireless communication

Hirsch

2016

J. Appl. Probab.

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Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

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“…To prove Theorem 1, we proceed in three steps, where we use the general method of global and local paths that has already been successfully applied in the literature; see [1], [2], [4], [13], and [35]. First, in Section 4.1 we discretise R d into boxes, allowing us to use results from percolation theory on lattices.…”

Section: Proof Of Theoremmentioning

confidence: 99%

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

Hirsch¹,

Neuhäuser²,

Gloaguen

et al. 2015

Advances in Applied Probability

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We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Cited by 12 publications

References 12 publications

Percolation for D2D networks on street systems

Percolation for D2D networks on street systems

Bounded-hop percolation and wireless communication

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

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