On the critical communication range under node placement with vanishing densities

We consider the random network where n points are placed independently on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some transmission range. When F admits a continuous density f with f_* = inf (f(x), x isin [0,1]) > 0, the property of graph connectivity for the underlying random graph is known to admit a strong critical threshold. Through a counterexample, we show that only a weak critical threshold exists when f_* = 0 and we identify it. Implications for the critical transmission range are discussed.