On the connectivity properties of wireless multi-hop networks

Given a multi-hop network in which a total of n nodes are randomly and independently distributed in a unit square following a uniform distribution and each node has a uniform transmission range r(n), and two distinct nodes can directly communicate with each other if and only if their Euclidean distance is at most r(n), this paper investigates the characteristics of the minimum transmission range r_c(n), at which the network is connected with a high probability. We show that for small values of n, r_c ²(n) grows approximately linearly with 1/n ; and as n goes to infinity, r_c ²(n) scales with log n/n. Simulations are performed to verify the theoretical analysis. The results of this paper are very useful in the design and dimensioning of wireless sensor networks and wireless ad hoc networks.