Very sharp transitions in one-dimensional MANETs

Abstract

We investigate how quickly phase transitions can occur in one-dimensional geometric random graph models of MANETs. In the case of graph connectivity, we show that the transition width behaves like 1/n (when the number n of users is large), a significant improvement over general asymptotic bounds given recently by Goel et al. for monotone graph properties. We also discuss a similar result for the property that there exists no isolated user in the network. The asymptotic results are validated by numerical computations. Finally we outline how the approach sed here could be applied in higher dimensions and or other graph properties.