In this paper, we study the well-known phase transition behavior of connectivity in a wireless multi-hop network, but, in contrast to other studies, in a shadowing environment. We consider that a total of n nodes are randomly, independently and uniformly distributed on a unit square in R2, each node has a uniform transmission power and any two nodes are directly connected if and only if the power received by one node from the other node, as determined by the log-normal shadowing model, is larger than or equal to a given threshold. We extend the results obtained under the unit disk communication model in previous work to the more realistic log-normal shadowing model, and derive an analytical formula for the phase transition width of connectivity for large n. We also demonstrate how our results can be extended to higher dimensional networks and to other channel models.

In this paper, we study the giant component, the largest component containing a non-vanishing fraction of nodes, in wireless multi-hop networks in Rd(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 analytical formula for calculating the probability of having a giant component of order above pn with any fixed 0.5 < p les 1. The asymptotic behavior of one dimensional network having a giant component is investigated based on the derived result, which is distinctly different from its two dimensional counterpart. For d = 2, we derive an asymptotic analytical upper bound on the minimum transmission range at which the probability of having a giant component of order above qn for any fixed 0 < q < 1 tends to one as n rarr infin. Based on the result, we show that significant energy savings can be achieved if we only require a large percentage of nodes (e.g. 95%) to be connected rather than requiring all nodes to be connected. The results of this paper are of practical significance in the design and analysis of wireless ad hoc networks and sensor networks.

In this paper, we study the giant component, the largest component containing a non-vanishing fraction of nodes, in a wireless multi-hop network where n nodes are randomly and uniformly distributed in [0, 1]d(d = 1, 2) and any two nodes can communicate directly with each other if their Euclidean distance is not larger than the transmission range r. We investigate the probability that the size of the giant component is at least a given threshold p with 0.5 < p les 1. For d = 1, we derive a closed-form analytical formula for this probability. For d = 2, we propose an empirical formula for this probability using simulations. In addition, we compare the transmission range required for having a connected network with the transmission range required for having a certain size giant component for d = 2. The comparison shows that a significant energy saving can be achieved if we only require most nodes (e.g. 95%) to be connected to the giant component rather than require all nodes to be connected. The results of this paper are of practical value in the design and analysis of wireless ad hoc networks and sensor networks.

In this paper, we study the phase transition behavior of k-connectivity (k=1,2,...) in wireless multihop networks where a total of n nodes are randomly and independently distributed following a uniform distribution in the unit cube [0,1]d(d = 1,2,3), and each node has a uniform transmission range r(n). It has been shown that the phase transition of k-connectivity becomes sharper as the total number of nodes n increases. In this paper, we investigate how fast such phase transition happens and derive a generic analytical formula for the phase transition width of k-connectivity for large enough n and for any fixed positive integer k in d-dimensional space by resorting to a Poisson approximation for the node placement. This result also applies to mobile networks where nodes always move randomly and independently. Our simulations show that to achieve a good accuracy, n should be larger than 200 when k = 1 and d = 1; and n should be larger than 600 when k les 3 and d = 2, 3. The results in this paper are important for understanding the phase transition phenomenon; and it also provides valuable insight into the design of wireless multihop networks and the understanding of its characteristics.

Consider a wireless multi-hop network formed by distributing a total of n nodes randomly and uniformly in the unit cube [0, 1]d(d = 1, 2, 3) and connecting any two distinct nodes directly iff (if and only if) their Euclidean distance is not greater than a given threshold r(n). We study the phase transition phenomenon of a k-connected (k isin Nopf) multi-hop network in this paper. We show that the phase transition of k-connectivity becomes sharper as n increases. We derive a generic analytical formula for the phase transition width for large n and for any fixed k isin Nopf in d-dimensional space. The result in this paper is important for understanding phase transition behavior, and it provides valuable insight into the design and implementation of wireless multi-hop networks.

Given a wireless sensor network (WSN) whose sensors are randomly and independently distributed in a bounded area following a homogeneous Poisson process with density rho and each sensor has a uniform transmission radius of r0, we investigate the probability that two random sensors separated by a known distance x are fc-hop neighbors for some positive integer k in this paper. We give a closed-form equation for computing this probability for k = 2; and also give a recursive equation for evaluating this probability for k > 2 by using some approximations. The accuracy of the approximate analytical solution is validated by simulations. Furthermore, we present an empirical method to correct the discrepancies between the analytical results and the simulation results caused by the approximation. The result of this paper can be useful in a number of sensor network problems, e.g., estimating the transmission delay between two sensors and energy consumed in the transmission, and WSN routing problems.