Threshold Functions, Node Isolation, and Emergent Lacunae in Sensor Networks

A geometrically random network of sensors is obtained by modeling sensors as random points in the unit disc equipped with a local sensing capability and the ability to communicate with other sensors in their vicinity. Node extinctions in the network representing the finite battery lifetimes of the sensors are modeled as a sequence of independent random variables governed by a common probability distribution parametrized by the sensing and communication radii of the sensor nodes. Following its establishment, the devolution of the network with time is characterized by the appearance first of isolated nodes, then the growth of sensory lacunae or dead spots in the sensor field, and, eventually, a breakdown in connectivity between survivors. It is shown that these phenomena occur very sharply in time, these phase transitions occurring at times characteristic of the underlying probability law governing lifetimes. More precisely, it is shown that as the number of sensors grows there exists a critical point in time determined solely by the lifetime distribution at which the number of emergent lacunae of a given size is asymptotically Poisson