| The complex nature of wireless ad hoc networks requires fundamental research to address issues related to their modeling, performance analysis, design, and implementation. In the first part of this dissertation, we study the topological properties of wireless networks, abstracting them as random geometric graphs. We study how many neighbors each individual node needs to maintain in order to have certain desirable global properties. In order to guarantee being connected with probability going to one as the network size goes to infinity, we show that each node needs to connect to its theta(log n) nearest neighbors. Specifically, 1.44logn neighbors can guarantee asymptotic connectivity, while 0.074logn leads to disconnectivity. We next study the theta-coverage of wireless networks of n nodes, with each node connecting to its &phis; n nearest neighbors. The network is called theta-covered if every node, except those along the boundary, can find an edge to one of its &phis;n nearest neighbors within any angle theta. Our result shows that &phis;n = log2p 2p-q n is the exact threshold function: as n goes to infinity, (1 + delta) log2p 2p-q n neighbors for each node leads to theta-coverage with probability approaching one as n → infinity, while (1 - delta) log2p 2p-q n does not, for any delta > 0.; In the second part of this dissertation, we study how fading, a fundamental aspect of all wireless communications, influences the scaling of the transport capacity of wireless networks. The transport capacity is the supremum of the sum of distance-rate product over all source destination pairs. Our results show that in the high attenuation regime when the absorption constant of the medium is positive or the path-loss exponent is larger than 3, the transport capacity scales as theta(n), whenever every pair of nodes is separated by at least a constant distance. This is true in various fading environments and for any causal strategy. This fundamental result shows that the current technological strategy of relaying by repeated decode-and-forward is order optimal even in the presence of fading. |
