Connectivity of Large Scale Networks: Distribution of Isolated Nodes
  Connectivity is one of the most fundamental properties of wireless multi-hop networks. A network is said to be connected if there is a path between any pair of nodes. A convenient way to study the connectivity of a random network is by investigating the condition under which the network has no isolated node. The condition under which the network has no isolated node provides a necessary condition for a connected network. Further the condition for a network to have no isolated node and the condition for the network to be connected can often be shown to asymptotically converge to be the same as the number of nodes approaches infinity, given a suitably defined random network and connection model. Currently analytical results on the distribution of the number of isolated nodes only exist for the unit disk model. This study advances research in the area by providing the asymptotic distribution of the number of isolated nodes in random networks with nodes Poissonly distributed on a unit square under a generic random connection model. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results, together with results in a companion paper on the sufficient condition for a network to be connected, expand recent results obtained for connectivity of random geometric graphs assuming a unit disk model to results assuming a more generic and more practical random connection model. 
Guoqiang Mao and Brian DO Anderson
cs.IT cs.NI math.IT