Extremal properties of three-dimensional sensor networks with applications

We analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions. For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (respectively, below) which the property exists with high (respectively, a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors, and their transmitting/sensing ranges. More specifically, we consider the following problems: assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3-dimensional region R of volume V, how large must the sensing range R_SENSE be to ensure a given degree of coverage of the region to monitor? For a given transmission range R_TRANS, what is the minimum (respectively, maximum) degree of the network? What is then the typical hop diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks.