Very sharp transitions in one-dimensional MANETs

We investigate how quickly phase transitions can occur in one-dimensional geometric random graph models of MANETs. In the case of graph connectivity and the absence of isolated nodes, we show that the transition width behaves like n-1 (when the number n of nodes is large), a significant improvement over asymptotic bounds given recently by Goel et al. for arbitrary monotone graph properties. We also outline how the approach used here could be applied in higher dimensions and for other graph properties.