On the Giant Component in Wireless Multi-Hop Networks

In this paper, we study the giant component, the largest component containing a non-vanishing fraction of nodes, in a wireless multi-hop network where n nodes are randomly and uniformly distributed in [0, 1]^d (d = 1, 2) and any two nodes can communicate directly with each other if their Euclidean distance is not larger than the transmission range r. We investigate the probability that the size of the giant component is at least a given threshold p with 0.5 < p les 1. For d = 1, we derive a closed-form analytical formula for this probability. For d = 2, we propose an empirical formula for this probability using simulations. In addition, we compare the transmission range required for having a connected network with the transmission range required for having a certain size giant component for d = 2. The comparison shows that a significant energy saving can be achieved if we only require most nodes (e.g. 95%) to be connected to the giant component rather than require all nodes to be connected. The results of this paper are of practical value in the design and analysis of wireless ad hoc networks and sensor networks.