A Theory of Routing for Large-Scale Wireless Ad-Hoc Networks
  In this work we develop a new theory to analyse the process of routing in large-scale ad-hoc wireless networks. We use a path integral formulation to examine the properties of the paths generated by different routing strategies in these kinds of networks. Using this theoretical framework, we calculate the statistical distribution of the distances between any source to any destination in the network, hence we are able to deduce a length parameter that is unique for each routing strategy. This parameter, defined as the {\it effective radius}, effectively encodes the routing information required by a node. Analysing the aforementioned statistical distribution for different routing strategies, we obtain a threefold result for practical Large-Scale Wireless Ad-Hoc Networks: 1) We obtain the distribution of the lengths of all the paths in a network for any given routing strategy, 2) We are able to identify "good" routing strategies depending on the evolution of its effective radius as the number of nodes, $N$, increases to infinity, 3) For any routing strategy with finite effective radius, we demonstrate that, in a large-scale network, is equivalent to a random routing strategy and that its transport capacity scales as $\Theta(\sqrt{N})$ bit-meters per second, thus retrieving the scaling law that Gupta and Kumar (2000) obtained as the limit for single-route large-scale wireless networks. 
Antonio J. Caama\~no, Juan J. Vinagre, Mark Wilby and Javier Ramos