On the Phase Transition Width of K-Connectivity in Wireless Multihop Networks

In this paper, we study the phase transition behavior of k-connectivity (k=1,2,...) in wireless multihop networks where a total of n nodes are randomly and independently distributed following a uniform distribution in the unit cube [0,1]^d (d = 1,2,3), and each node has a uniform transmission range r(n). It has been shown that the phase transition of k-connectivity becomes sharper as the total number of nodes n increases. In this paper, we investigate how fast such phase transition happens and derive a generic analytical formula for the phase transition width of k-connectivity for large enough n and for any fixed positive integer k in d-dimensional space by resorting to a Poisson approximation for the node placement. This result also applies to mobile networks where nodes always move randomly and independently. Our simulations show that to achieve a good accuracy, n should be larger than 200 when k = 1 and d = 1; and n should be larger than 600 when k les 3 and d = 2, 3. The results in this paper are important for understanding the phase transition phenomenon; and it also provides valuable insight into the design of wireless multihop networks and the understanding of its characteristics.