Sharp thresholds for relative neighborhood graphs in wireless Ad Hoc networks
Chih‐Wei Yi, Peng‐Jun Wan, Lixin Wang, Chao-Min Su · IEEE Transactions on Wireless Communications · 2010
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 > ß0and at least ß ¿(1n n/¿n) for any fixed 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-¿.