Sharp thresholds for relative neighborhood graphs in wireless Ad Hoc networks

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 ¿ in a unit-area disk. Let ß₀ = ¿(1/(2/3 - ¿(3)/2¿)) ¿ 1.6. We show that the largest RNG edge length is asymptotically almost surely at most ß ¿(1n n/¿n) for any fixed ß > ß₀ and at least ß ¿(1n n/¿n) for any fixed ß < ß₀. This implies that the threshold width of the critical transmission radius is o (¿(1n n/n)). In addition, we also prove that for any constant ¿, the expected number of RNG edges whose lengths are not less than ß₀ ¿(1n n+¿)/¿n is asymptotically equal to ß0²/2 e^-¿.