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## Percolation in multi-hop wireless networks ¤ (2003)

### Citations

3329 |
Collective dynamics of ’small-world’ networks
- Watts, Strogatz
- 1998
(Show Context)
Citation Context ...ly, the idea that some longer connections help to reach percolation at a lower density of points is also related to the small world networks described, for example, in the paper by Watts and Strogatz =-=[17]-=-. 2.2 Shifting and squeezing Another transformation of g that we consider is gshifts (x). Here we `shift' the function g outwards (so that a disc becomes an annulus, for example) by a distance s, but ... |

3242 | The capacity of wireless networks
- Gupta, Kumar
- 1998
(Show Context)
Citation Context ...been a growing interest in modelling global properties of these networks as a function of the node population, and basic results on connectivity and throughput capacity have been obtained [3] [5] [6] =-=[7]-=- [9]. These works typically assume that nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited these ideas a... |

1220 | Mobility Increases the Capacity of Adhoc Wireless Networks,”
- Grossglauser, Tse
- 2002
(Show Context)
Citation Context ...ere has been a growing interest in modelling global properties of these networks as a function of the node population, and basic results on connectivity and throughput capacity have been obtained [3] =-=[5]-=- [6] [7] [9]. These works typically assume that nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited these... |

541 | Critical power for asymptotic connectivity in wireless networks,
- Gupta, Kumar
- 1998
(Show Context)
Citation Context ...has been a growing interest in modelling global properties of these networks as a function of the node population, and basic results on connectivity and throughput capacity have been obtained [3] [5] =-=[6]-=- [7] [9]. These works typically assume that nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited these ide... |

325 |
Continuum Percolation,
- Meester, Roy
- 1996
(Show Context)
Citation Context ...s appeared in the 1961 paper of Gilbert's [4], who studied the problem of multi-hop connectivity of wireless broadcasting stations, establishing the foundations of the theory of Continuum Percolation =-=[11]-=-. In his formulation, points of a two-dimensional Poisson point process represent wireless transmitting stations of range 2r and he asks if the system can provide some long-distance communication. He ... |

186 |
Random plane networks.
- Gilbert
- 1961
(Show Context)
Citation Context ... nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited these ideas appeared in the 1961 paper of Gilbert's =-=[4]-=-, who studied the problem of multi-hop connectivity of wireless broadcasting stations, establishing the foundations of the theory of Continuum Percolation [11]. In his formulation, points of a two-dim... |

148 | A deterministic approach to throughput scaling in wireless networks”,
- Kulkarni, Viswanath
- 2004
(Show Context)
Citation Context ... a growing interest in modelling global properties of these networks as a function of the node population, and basic results on connectivity and throughput capacity have been obtained [3] [5] [6] [7] =-=[9]-=-. These works typically assume that nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited these ideas appea... |

48 | Covering algorithms, continuum percolation, and the geometry of wireless networks
- Booth, Bruck, et al.
(Show Context)
Citation Context ... possible. In the past forty years Gilbert's model has received much attention from mathematicians and physicists and, recently, extensions relevant to wireless network design have also been proposed =-=[2]-=- [16]. In this paper we consider a further extension that is useful to model more realistic, unreliable and non-rotationally symmetric communication channels. Our motivation is made clear by looking a... |

42 | Efficient measurement of the percolation threshold for fully penetrable discs
- Quintanilla, Torquato, et al.
- 2000
(Show Context)
Citation Context ...e, the solid disc at the top is at a height of around 4.5, indicating that disc percolation occurs whenever each disc has over 4.5 neighbors on average, in agreement with numerical results such as in =-=[14]-=-. The bottom of the graph is at ENC of 1, which is the theoretical minimum, attained by very large, di®use shapes [13]. The ascending shapes on the left are discs with probabilistic connections to oth... |

15 |
On the spread-out limit for bond and continuum percolation.
- Penrose
- 1992
(Show Context)
Citation Context ...Another way of looking at this is that the longer links introduced by stretching the connection function are making up for the increased unreliability of the connections. In some related work Penrose =-=[13]-=- has shown that as a connection function of e®ective area 1 gets more spread out, its critical density for percolation converges to 1. This can be seen as the limiting case of our Theorem 2.1. Meester... |

14 | Designing a contact process: the piecewise-homogeneous process on a finite set with applications
- Wagner, Anantharam
- 2005
(Show Context)
Citation Context ...sible. In the past forty years Gilbert's model has received much attention from mathematicians and physicists and, recently, extensions relevant to wireless network design have also been proposed [2] =-=[16]-=-. In this paper we consider a further extension that is useful to model more realistic, unreliable and non-rotationally symmetric communication channels. Our motivation is made clear by looking at Fig... |

12 |
Continuum percolation threshold for interpenetrating squares and cubes.
- Baker, Paul, et al.
- 2002
(Show Context)
Citation Context ...fect links by considering a random connection model where each pair of points (xi; xj) of a Poisson point process of density ¸ is connected with probability g(xi¡xj), for some given function g : R2 ! =-=[0; 1]-=-. We may, for example, pick a function g such that the probability of existence of a link between a transmitter and a receiver decreases as the two points get further away. For generality, however, we... |

9 | Optimization of shape in continuum percolation
- Jonasson
(Show Context)
Citation Context ...irection overlap each other. If a node falls inside the shape of another node, then the reverse is true as well. This property does not hold for non-centrally symmetric shapes, see Figure 5. Jonasson =-=[10]-=- has shown that if any convex shape of area 1 percolates at certain density, then triangles (that are not centrally symmetric) of area 1 will also do so. Roy and Tanemura [15] strengthened this result... |

7 | Nearest neighbour and hard sphere models in continuum percolation. Random Structures Algorithms 9 - HÄGGSTRÖM, MEESTER - 1996 |

7 |
Critical intensities of boolean models with different underlying convex shapes
- Roy, Tanemura
(Show Context)
Citation Context ..., see Figure 5. Jonasson [10] has shown that if any convex shape of area 1 percolates at certain density, then triangles (that are not centrally symmetric) of area 1 will also do so. Roy and Tanemura =-=[15]-=- strengthened this result to show a strict inequality between the critical density of triangles and that of any other given convex shape of the same area. Jonasson [10] has also shown that the convex ... |

4 |
Thiran (2003). Impact of interferences on connectivity in ad-hoc networks
- Dousse, Baccelli, et al.
(Show Context)
Citation Context ..., there has been a growing interest in modelling global properties of these networks as a function of the node population, and basic results on connectivity and throughput capacity have been obtained =-=[3]-=- [5] [6] [7] [9]. These works typically assume that nodes are randomly located and that they are able to reliably relay messages to su±ciently close neighbors. A ¯rst stochastic model that exploited t... |

4 |
The random connection model in high dimensions
- Meester, Penrose, et al.
- 1997
(Show Context)
Citation Context ...onnection function of e®ective area 1 gets more spread out, its critical density for percolation converges to 1. This can be seen as the limiting case of our Theorem 2.1. Meester, Penrose, and Sarkar =-=[12]-=- proved a similar result as the dimension tends to in¯nity. Philosophically, the idea that some longer connections help to reach percolation at a lower density of points is also related to the small w... |