On the Absence of Isolated Nodes in Wireless Ad-Hoc Networks with Unreliable Links - a Curious Gap

We consider an extension to the disk model in one dimension where communication links established between nodes may fail. With the help of the method of first and second moments, we investigate the zero-one laws for the property that there are no isolated nodes in the underlying random graph. Two specific situations are discussed: For the unit circle we prove a full zero-one law and determine its critical scaling. For the unit interval we derive a zero-law and a one-law which capture deviations from different critical scalings; a completely symmetric zero-one law is established under an additional condition. Analysis and simulations both indicate the possible presence of a gap between the one-law critical scalings for the unit interval and the unit circle. This discrepancy is quite surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on the unit interval and on the unit circle. Connections to recent results by Yi et al. are discussed.