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## On the giant component of wireless multi-hop networks in the presence of shadowing (2009)

Venue: | IEEE Transactions on Vehicular Technology |

Citations: | 8 - 4 self |

### Citations

2380 | Random graphs.
- Bollobas
- 1998
(Show Context)
Citation Context ...iant component has been extensively investigated in the literature for Bernoulli random graphs, and an analytical formula relating the giant component size and the average node degree2 has been found =-=[7]-=-. However, the Bernoulli random graph is not suitable for modeling wireless multi-hop networks, hence, it is inappropriate to apply the results on the giant component size obtained from Bernoulli rand... |

538 | Critical power for asymptotic connectivity in wireless networks
- Gupta, PR
- 1998
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Citation Context ...t, the largest component containing a non-vanishing fraction of nodes, in wireless multi-hop networks in <d (d = 1, 2). We assume that n nodes are randomly, independently and uniformly distributed in =-=[0, 1]-=-d, and each node has a uniform transmission range of r = r(n) and any two nodes can communicate directly with each other iff their Euclidean distance is at most r. For d = 1, we derive a closed-form a... |

323 |
Continuum Percolation
- Meester, Roy
- 1996
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Citation Context ...is different from higher-dimensional networks (e.g. d = 2). As we shall see, for d ≥ 2, if nrd →∞ as n→∞, almost surely the network only consists of isolated nodes and a unique giant component as n→∞ =-=[12]-=-. In addition, when the last isolated node vanishes, the network becomes connected almost surely [1], [10]. However, for d = 1, there may be multiple giant components (Theorem 2); and when the last is... |

320 | The number of neighbors needed for connectivity of wireless networks
- Xue, Kumar
(Show Context)
Citation Context ...rk are parts of a single component. In the past several years, the connectivity problem in wireless multi-hop networks has been widely investigated and significant outcome has been achieved [1], [2], =-=[3]-=-. However, from a practical point of view, requiring all nodes to be connected may be a too stringent condition to satisfy. It has been shown by simulations that the transmission range required for a ... |

317 | On the minimum node degree and connectivity of a wireless multihop network,” MOBIHOC - Bettstetter - 2002 |

37 | Connectivity in Wireless Ad Hoc Networks with a Log-normal Radio Model
- Hekmat, Mieghem
- 2006
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Citation Context ...isfy. It has been shown by simulations that the transmission range required for a large percentage of nodes to be connected is much less than the transmission range for all nodes to be connected [4], =-=[5]-=-, [6]. Fig. 1 shows the average value of the ratio of the transmission range required for 95% of nodes to be connected to the transmission range required for a connected network. As shown in the figur... |

16 |
On the connectivity of a random interval graph. Random Structures & Algorithms
- Godehardt, Jaworski
- 1998
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Citation Context ...−1)j(1− (j + 2)r)n + min{n−1,b 1r c}∑ j=0 ( n− 1 j ) (−1)j(1− jr)n. (1) In order to prove Theorem 1, the following three lemmas, viz., Lemma 1, Lemma 2 and Lemma 3, are needed. Lemma 1 (Lemma 1 in =-=[11]-=-). Let [x, x+y] be a subinterval of length y on a unit interval [0, 1]. Let two of k given vertices be 3placed on the borders of this subinterval. Let P (k, y, r) be the probability that the remaining... |

10 | Giant Clusters in Random Ad Hoc Networks
- Németh, Vattay
- 2003
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Citation Context ...d on the analytical results obtained in Bernoulli random graphs, the authors proposed an empirical formula relating the giant component size and the average node degree in random geometric graphs. In =-=[8]-=-, Németh et al. investigated the giant component size by using a fractal propagation model where the probability of having a link between two nodes is determined by their Euclidean distance and two n... |

3 |
Random Geometric Graphs, 1st ed
- Penrose
- 2003
(Show Context)
Citation Context ...quely represents a node and each edge of the set E uniquely represents a wireless link, and vice versa. In this paper, we model wireless multi-hop networks by widely used random geometric graphs [1], =-=[10]-=-. Typically, a random geometric graph is defined as follows: Definition 1 ( [10]). Let X1, X2, ..., Xn be n points which are independently, randomly and uniformly distributed in [0, 1]d (d = 1, 2); le... |

3 |
Poisson convergence can yield very sharp transitions in geometric random graphs
- Han, Makowski
- 2006
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Citation Context ...lowing intuitive explanation for the result. It is well known that the width of the phase transition region from an almost disconnected network to an almost connected network approaches zero as n → ∞ =-=[13]-=-. This means at large n, the probability of having a connected network as a function of r is almost like a step function such that at a certain value of r (termed the critical transmission range), a t... |

2 |
Phase Transition and Connectivity
- Raghavan, Thadakamalla, et al.
- 2005
(Show Context)
Citation Context ... It has been shown by simulations that the transmission range required for a large percentage of nodes to be connected is much less than the transmission range for all nodes to be connected [4], [5], =-=[6]-=-. Fig. 1 shows the average value of the ratio of the transmission range required for 95% of nodes to be connected to the transmission range required for a connected network. As shown in the figure, wh... |

1 |
The Critical Transimitting Range for Connectivity in Sparse Wireless Ad Hoc Networks
- Santi, Blough
- 2003
(Show Context)
Citation Context ...o satisfy. It has been shown by simulations that the transmission range required for a large percentage of nodes to be connected is much less than the transmission range for all nodes to be connected =-=[4]-=-, [5], [6]. Fig. 1 shows the average value of the ratio of the transmission range required for 95% of nodes to be connected to the transmission range required for a connected network. As shown in the ... |

1 |
Giant Component and Connectivity
- Bradonjić, Hagberg, et al.
- 2007
(Show Context)
Citation Context ...mula for the critical transmission range at which the network has a giant component with a high probability, and they showed that the critical range is approximately inversely proportional to √ n. In =-=[9]-=-, Bradonjić et al. studied the giant component using a network model based on a geographical threshold graph which is almost the same as the random geometric graph except that the link existence betw... |