In this chapter, we discuss models for wireless multihop networks that generalize the Gaussian channel models we studied earlier. We extend the cutset bound and the noisy network coding inner bound on the capacity region of the multimessage DMN presented in Chapter 18 to Gaussian networks. We show through a Gaussian two-way relay channel example that noisy network coding can outperform decode–forward and amplify– forward, achieving rates within a constant gap of the cutset bound while the inner bounds achieved by these other schemes can have an arbitrarily large gap to the cutset bound. More generally, we show that noisy network coding for the Gaussian multimessage multicast network achieves rates within a constant gap of the capacity region independent of network topology and channel gains. For Gaussian networks with other messaging demands, e.g., general multiple-unicast networks, however, no such constant gap results exist in general. Can we still obtain some guarantees on the capacity of these networks?

To address this question, we introduce the scaling-law approach to capacity, where we seek to find the order of capacity scaling as the number of nodes in the network becomes large. In addition to providing some guarantees on network capacity, the study of capacity scaling sheds light on the role of cooperation through relaying in combating interference and path loss in large wireless networks. We first illustrate the scaling-law approach via a simple unicast network example that shows how relaying can dramatically increase the capacity by reducing the effect of high path loss. We then present the Gupta–Kumar random network model in which the nodes are randomly distributed over a geographical area and the goal is to determine the capacity scaling law that holds for most such networks. We establish lower and upper bounds on the capacity scaling law for the multiple-unicast case. The lower bound is achieved via a cellular time-division scheme in which the messages are sent simultaneously using a simple multihop scheme with nodes in cells along the lines from each source to its destination acting as relays. We show that this scheme achieves much higher rates than direct transmission with time division, which demonstrates the role of relaying in mitigating interference in large networks.