The longest edge of the random minimal spanning tree

Penrose, Mathew D.

doi:10.1214/aoap/1034625335

Cited by 420 publications

(369 citation statements)

References 31 publications

(49 reference statements)

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“…Under the given deployment area of a sensor network, increasing the number of nodes (N) or the communication range (r) will respectively increase the number of connections in the network. To obtain the appropriate value of r which connects N sensor nodes with the desired level of connectivity, we utilize the results from (Penrose, 1997). For N points placed uniformly at random on the unit square in the 2-dimensional space, Penrose (Penrose, 1997) found an asymptotic bound on the length of the longest edge (M n ) of MST (Minimum Spanning Tree) as follows lim…”

Section: Simulation Resultsmentioning

confidence: 99%

Group Key Managements in Wireless Sensor Networks

Son¹,

Seo²

2010

Sustainable Wireless Sensor Networks

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A sensor network operating in open environments requires a network-wide group key for confidentiality of exchanged messages between sensor nodes. When a node behaves abnormally due to its malfunction or a compromise attack by adversaries, the central sink node should update the group key of other nodes. The major concern of this group key update procedure will be the multi-hop communication overheads of the rekeying messages due to the energy constraints of sensor nodes. Many researchers have tried to reduce the number of rekeying messages by using the logical key tree. In this chapter, we propose an energy-efficient group key management scheme called Topological Key Hierarchy (TKH). TKH generates a key tree by using the underlying sensor network topology with consideration of subtree-based key tree separation and wireless multicast advantage. Based on our detailed analysis and simulation study, we compare the total rekeying costs of our scheme with the previous logical key tree schemes and demonstrate its energy efficiency.

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Section: Simulation Resultsmentioning

confidence: 99%

Group Key Managements in Wireless Sensor Networks

Son¹,

Seo²

2010

Sustainable Wireless Sensor Networks

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“…Related problems to graph theoretic results in this paper have been studied in the context of random graph theory [5], continuum percolation and geometric probability [6][7][8][9][10] and the study of wireless network graphs [11][12][13][14][15][16]. In random graph theory, the model G(n, p) is extensively studied, in which edges appear in a graph of n vertices with probability p independently of each other.…”

Section: Related Workmentioning

confidence: 99%

Connectivity properties of large-scale sensor networks

2009

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In wireless sensor networks, both nodes and links are prone to failures. In this paper we study connectivity properties of large-scale wireless sensor networks and discuss their implicit effect on routing algorithms and network reliability. We assume a network model of n sensors which are distributed randomly over a field based on a given distribution function. The sensors may be unreliable with a probability distribution, which possibly depends on n and the location of sensors. Two active sensor nodes are connected with probability p e (n) if they are within communication range of each other. We prove a general result relating unreliable sensor networks to reliable networks. We investigate different graph theoretic properties of sensor networks such as k-connectivity and the existence of the giant component. While connectivity (i.e. k = 1) insures that all nodes can communicate with each other, k-connectivity for k [ 1 is required for multipath routing. We analyze the average shortest path of the k paths from a node in the sensing field back to a base station. It is found that the lengths of these multiple paths in a k-connected network are all close to the shortest path.These results are shown through graph theoretical derivations and are also verified through simulations.

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“…We use some results from continuum percolation by Penrose [13] [14]. Consider a graph where nodes are distributed as a homogeneous Poisson process in R 2 with intensity λ.…”

Section: For All N > N( θ C)mentioning

confidence: 99%

Asymptotic Connectivity in Wireless Networks Using Directional Antennas

Zhang

Fang

2007

27th International Conference on Distributed Computing Systems (ICDCS '07)

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Connectivity is a crucial issue in wireless networks. Gupta and Kumar show that with omnidirectional antennas, the critical transmission range for a wireless network to achieve asymptotic connectivity is O( log n n ) if n nodes are uniformly and independently distributed in a disk of unit area. In this paper, we investigate the connectivity problem when directional antennas are used. We find that there also exists a critical transmission range, which corresponds to a critical transmission power. We show that in the same propagation environment, when directional antennas use the optimal antenna pattern, the critical transmission power could be much smaller than that in networks using omnidirectional antennas. Moreover, to achieve asymptotic connectivity, it is known that each node has to have O(log n) neighbors when using omnidirectional antennas. We show that even using the transmission power level at which each node has only O(1) neighbors when using omnidirectional antennas, we can still achieve the asymptotic connectivity with directional antennas.

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The longest edge of the random minimal spanning tree

Cited by 420 publications

References 31 publications

Group Key Managements in Wireless Sensor Networks

Group Key Managements in Wireless Sensor Networks

Connectivity properties of large-scale sensor networks

Asymptotic Connectivity in Wireless Networks Using Directional Antennas

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