Sharp Thresholds for Relative Neighborhood Graphs in Wireless Ad Hoc Networks

Abstract

In wireless ad hoc networks, relative neighborhood graphs (RNGs) are widely used for topology control. If every node has the same transmission radius, then an RNG can be locally constructed by using only one hop information if the transmission radius is set no less than the largest edge length of the RNG. The largest RNG edge length is called the critical transmission radius for the RNG. In this paper, we consider the RNG over a Poisson point process with mean density n in a unit-area disk. Let. beta(0) = root 1/(2/3 - root 3/2 pi) approximate to 1.6. We show that the largest RNG edge length is asymptotically almost surely at most beta root ln n/pi n for any fixed beta > beta(0) and at least beta root ln n/pi n for any fixed beta > beta(0). This implies that the threshold width of the critical transmission radius is o(root ln n/n). In addition, we also prove that for any constant xi, the expected number of RNG edges whose lengths are not less than beta(0)root ln n+xi/pi n is asymptotically equal to beta(2)(0)/2 e(-xi).