Connectivity in Sensor Networks

We analyze the probability that an arbitrarily selected sensor node in a sensor network is connected to a specified number, m, of other sensor nodes; i.e., that a communication path exists between a selected node and at least m other nodes. Specifically, we consider the following problem: If given a collection of sensor nodes, each with a communication radius r, that are to be uniformly distributed over an area of size A, how many nodes must be deployed to assure that any node is connected to at least m other nodes with a specified probability? Alternatively, if we deploy a fixed number of nodes over a region, what is the probability that a selected node will have m neighbors? Such questions are particularly pertinent for unattended sensor networks that perform cooperative operations requiring the participation of at least m nodes