We investigate the condition on transmission radius needed to achieveconnectivity in duty-cycled wireless sensor networks (briefly, DC-WSN). First,we settle a conjecture of Das et. al. (2012) and prove that the connectivitycondition on Random Geometric Graphs (RGG), given by Gupta and Kumar (1989),can be used to derive a weak sufficient condition to achieve connectivity inDC-WSN. To find a stronger result, we define a new vertex-based randomconnection model which is of independent interest. Following a proof techniqueof Penrose (1991) we prove that when the density of the nodes approachesinfinity then a finite component of size greater than 1 exists with probability0 in this model. We use this result to obtain an optimal condition on nodetransmission radius which is both necessary and sufficient to achieveconnectivity and is hence optimal. The optimality of such a radius is alsotested via simulation for two specific duty-cycle schemes, called thecontiguous and the random selection duty-cycle scheme. Finally, we design aminimum-radius duty-cycling scheme that achieves connectivity with atransmission radius arbitrarily close to the one required in Random GeometricGraphs. The overhead in this case is that we have to spend some time computingthe schedule.
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