Wireless Networking
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We consider the clustering properties of one-dimensional sensor networks where the nodes are randomly deployed. Unlike most other work on randomly deployed networks, we assume that the node locations are drawn from a non uniform distribution. Specifically, we consider the exponential distribution. We first obtain the probability that there exists a path between two labeled nodes in a randomly deployed network and obtain the limiting behavior of this probability. The probability mass function (pmf) for the number of components in the network is then obtained. We show that the number of components in the network converges in distribution. We also derive the probabilities for different locations of the components. We then obtain the probability for the existence of a k-sized component and components of size 驴 k. Asymptotics in the number of nodes in the network are computed for these probabilities. An interesting result that we obtain is that as the number of nodes in the network tends to infinity, a giant component in which a specific fraction 驴 of the nodes form a component almost surely does not exist for any 0 n驴, the network almost surely does not have a giant component.