On the Giant Component of Wireless Multi-hop Networks in the Presence of Shadowing

In this paper, we study transmission power to secure the connectivity of a network. Instead of requiring all nodes to be connected, we require that only a large fraction (e.g., 95%) be connected, which is called the giant component. We show that, with this slightly relaxed requirement on connectivity, significant energy savings can be achieved for a large-scale network. In particular, we assume that a total of n nodes are randomly independently uniformly distributed in a unit square in ℛ², that each node has uniform transmission power, and that any two nodes are directly connected if and only if the power that was received by one node from the other node, as determined by the log-normal shadowing model, is larger than or equal to a given threshold. First, we derive an upper bound on the minimum transmission power at which the probability of having a giant component of order above qn for any fixed q ∈ (0, 1) tends to one as n →∞. Second, we derive a lower bound on the minimum transmission power at which the probability of having a connected network tends to one as n →∞. We then show that the ratio of the aforementioned transmission power that was required for a giant component to the transmission power that was required for a connected network tends to zero as n →∞. This result implies significant energy savings if we require that only most nodes (e.g., 95%) be connected rather than requiring all nodes to be connected. This result is also applicable for any other arbitrary channel model that satisfies certain intuitively reasonable conditions.