Connectivity and capacity are two fundamental properties of wireless multi-hop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. the dense network model, the extended network model and the infinite network model, which consider respectively a network deployed in a fixed finite area with a sufficiently large node density, a network deployed in a sufficiently large area with a fixed node density, and a network deployed in @math with a sufficiently large node density. The infinite network model originated from continuum percolation theory and asymptotic results obtained from the infinite network model have often been applied to the dense and extended networks. In this paper, through two case studies related to network connectivity on the expected number of isolated nodes and on the vanishing of components of finite order k>1 respectively, we demonstrate some subtle but important differences between the infinite network model and the dense and extended network models. Therefore extra scrutiny has to be used in order for the results obtained from the infinite network model to be applicable to the dense and extended network models. Asymptotic results are also obtained on the expected number of isolated nodes, the vanishingly small impact of the boundary effect on the number of isolated nodes and the vanishing of components of finite order k>1 in the dense and extended network models using a generic random connection model.
Other work in the area include @cite_31 @cite_27 @cite_24 @cite_7 , which studies from the percolation perspective, the impact of mutual interference caused by simultaneous transmissions, the impact of physical layer cooperative transmissions, the impact of directional antennas and the impact of unreliable links on connectivity respectively.
@cite_24 @cite_27 @cite_31 @cite_7
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Connectivity is a crucial issue in wireless ad hoc networks (WANETs). Gupta and Kumar have shown that in WANETs using omnidirectional antennas, the critical transmission range to achieve asymptotic connectivity is O(radic(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 first assume that each node in the network randomly beam forms in one beam direction. We find that there also exists a critical transmission range for a WANET to achieve asymptotic connectivity, which corresponds to a critical transmission power (CTP). Since CTP is dependent on the directional antenna pattern, the number of beams, and the propagation environment, we then formulate a non-linear programming problem to minimize the CTP. We show that when directional antennas use the optimal antenna pattern, the CTP in a WANET using directional antennas at both transmitter and receiver is smaller than that when either transmitter or receiver uses directional antenna and is further smaller than that when only omnidirectional antennas are used. Moreover, we revisit the connectivity problem assuming that two neighboring nodes using directional antennas can be guaranteed to beam form to each other to carry out the transmission. A smaller critical transmission range than that in the previous case is found, which implies smaller CTP. Extensive research has demonstrated the potential improvement in physical layer performance when multiple radios transmit concurrently in the same radio channel. We consider how such cooperation affects the requirements for full connectivity and percolation in large wireless ad hoc networks. Both noncoherent and coherent cooperative transmission are considered. For one-dimensional (1-D) extended networks, in contrast to noncooperative networks, for any path loss exponent less than or equal to one, full connectivity occurs under the noncoherent cooperation model with probability one for any node density. Conversely, there is no full connectivity with probability one when the path loss exponent exceeds one, and the network does not percolate for any node density if the path loss exponent exceeds two. In two-dimensional (2-D) extended networks with noncoherent cooperation, for any path loss exponent less than or equal to two, full connectivity is achieved for any node density. Conversely, there is no full connectivity when the path loss exponent exceeds two, but the cooperative network percolates for node densities above a threshold which is strictly less than that of the noncooperative network. A less conclusive set of results is presented for the coherent case. Hence, even relatively simple noncoherent cooperation improves the connectivity of large ad hoc networks. We study connectivity and transmission latency in wireless networks with unreliable links from a percolation-based perspective. We first examine static models, where each link of the network is functional (active) with some probability, independently of all other links, where the probability may depend on the distance between the two nodes. We obtain analytical upper and lower bounds on the critical density for phase transition in this model. We then examine dynamic models, where each link is active or inactive according to a Markov on- off process. We show that a phase transition also exists in such dynamic networks, and the critical density for this model is the same as the one for static networks under some mild conditions. Furthermore, due to the dynamic behavior of links, a delay is incurred for any transmission even when propagation delay is ignored. We study the behavior of this transmission delay and show that the delay scales linearly with the Euclidean distance between the sender and the receiver when the network is in the subcritical phase, and the delay scales sub-linearly with the distance if the network is in the supercritical phase.