We consider N points 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 given transmission range R > 0. We define the critical transmission range R_N as the smallest transmission range such that the N nodes form a connected graph (under the notion of adjacency implied by the ability of nodes to communicate). The distribution of R_N being untractable, we are interested in developing an asymptotic theory for R_N as N becomes large. We carry out the discussion under the assumption that F admits a continuous density f. We identify two qualitatively different cases, namely f_min > 0 and f_min = 0 with f_min the minimum of f over [0,1]. In the process we make contact with the existence and nature of critical thresholds for the property of graph connectivity in the underlying geometric random graph. This is joint work with Ph.D. student Guang Han.





