# Topologi control in wireles ad hoc and sensor network - Santi P.

ISBN-10 0-470-09453-2

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By means of extensive simulation, the authors of (Santi and Blough 2003) argue that f(l) = log l is also a necessary condition for a.a.s. connectivity. We then claim the following result, which is only partially proven.

Proposition 4.2.5 If R = [0, l]d, with d = 2, 3, and n nodes are distributed uniformly at random in R, the CTR for connectivity is

ld log l rc = k ,

n

where k is a constant with 0 < k < 2ddd/2+1.

Let us finally comment about the giant component phenomenon in sparse ad hoc networks. Through simulations, it is observed in (Santi and Blough 2003) that the giant component phenomenon occurs in two- and three-dimensional networks, while it does not occur when nodes are located on a line. Although there is no formal proof of this fact, we can then conclude that sparse and dense ad hoc networks display the same behavior regarding the occurrence of the giant component.

4.3 The CTR with Different Deployment Region and Node Distribution

Characterizations of the CTR similar to those stated in the previous sections have been derived for different shapes of the deployment region R, and for different node distributions. In particular, Gupta and Kumar proved the same exact result as Corollary 4.1.2 when R is the disk of unit area (Gupta and Kumar 1998). The proof of Gupta and Kumar’s result is based on the theory of continuum percolation (see Appendix B), which is another important applied probability theory used in the analysis of ad hoc network properties.

Other authors considered the case in which nodes are distributed according to a Poisson process of a given intensity X. The CTR for Poisson distributed points on a line of length l is derived in (Piret 1991). A similar derivation of the CTR is obtained in (Dousse et al. 2002) when nodes are Poisson distributed on an unbounded one-dimensional region.

One observation regarding Poisson distribution is in order. With this type of distribution, the total number of deployed nodes is a random variable itself. In other words, with Poisson distribution, one is allowed to choose only the expected number of deployed nodes. For instance, if a Poisson process of intensity X is used to distribute nodes on a line of length I, an average of IX nodes will be deployed. So, setting X = f generates a network with n nodes on the average. Given this observation, Poisson distribution is used whenever the exact number of network nodes is not known, but some information on the expected node density is available to the network designer.

Another distribution that has been considered in the literature is the Normal distribution. This distribution models those situations in which nodes are somewhat concentrated around a certain point. For instance, if an ad hoc network is used to provide wireless Internet access, it is reasonable to assume that nodes are concentrated around the access point. Another example in which assuming Normally distributed nodes is reasonable is when wireless sensors are deployed in groups using a vehicle (e.g. a helicopter): in this situation,

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THE CTR FOR CONNECTIVITY: STATIONARY NETWORKS

node concentration around the release point is expected. The characterization of the CTR for connectivity in two- and three-dimensional networks with Normally distributed points is derived in (Penrose 1998).

Finally, we want to mention a recent result due to Penrose (Penrose 1999c), which characterizes the CTR for connectivity in case of arbitrary node distribution (provided certain technical conditions are satisfied). This important result, which is used in Chapter 5 to study the CTR in mobile ad hoc networks, essentially states that what determines the asymptotic behavior of the CTR is the minimal value of the probability density function F used to distribute nodes in the deployment region R.

4.4 Irregular Radio Coverage Area

As discussed in Section 2.2, the main limitation of the point graph model used to derive the results presented in this chapter is the assumption of regular radio coverage: for instance, in case of two-dimensional networks, the radio coverage region is assumed to be a disk of a certain radius centered at the transmitter. Given this weakness in the model, one might argue that the characterizations of the CTR introduced in the literature have scarce practical relevance. For this reason, some authors have recently investigated the conditions for a.a.s. connectivity in the presence of irregular radio coverage area. In this section, we discuss some interesting results presented in (Booth et al. 2003) and (Bettstetter 2004), which refer to two-dimensional ad hoc networks.

Consider a set of nodes located in the plane, and assume that nodes u and v are directly connected with a certain probability g(S(u, v)), where S(u, v) is the distance between the two nodes. Typically, g is a decreasing function of the distance. However, this is not imposed in the model, which allows g to be an arbitrary function of the distance.

Since the radio connectivity is defined in probabilistic terms, the model above allows irregular radio coverage area. For instance, there could exist nodes u,v,w such that S(u, v) < S(u, w), but only link (u, w) exists in the communication graph (see Figure 4.6). However, since the probability of having a link depends only on the distance between the nodes, the model can only represent situations in which the radio coverage area is rotary symmetric. For this reason, we call this model the rotary symmetric connection model.

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