The Critical Neighbourhood Range for Asymptotic Overlay Connectivity in Dense Ad Hoc Networks

We define, for an overlay built on top of an ad hoc network, a simple criterion for neighbourhood: two overlay nodes are neighbours if and only if there exists a path between them of at most R hops, and R is called the (overlay) neighbourhood range. A small R may result in a disconnected overlay, while an unnecessarily large R would generate extra control traffic. We are interested in the minimum R ensuring overlay connectivity, the so-called critical R. We derive a necessary and sufficient condition on R to achieve asymptotic connectivity of the overlay almost surely, i.e. connectivity with probability 1 when the number of overlay nodes tends to infinity, under the hypothesis that the underlying ad hoc network is itself asymptotically almost surely connected. This condition, though asymptotic, sheds some light on the relation linking the critical R to the number of nodes n, the normalized radio transmission range r and the overlay density D (i.e., the proportion of overlay nodes). This condition can be considered as an approximation when the number of nodes is large enough. Since r is considered as a function of n, we are able to study the impact of topology control mechanisms, by showing how the shape of this function impacts the critical R.