Design optimization of multi-sink sensor networks by analogy to electrostatic theory

In this work we introduce a new mathematical tool for optimization of routes, and topology design in wireless sensor networks. We introduce a vector field formulation that models communication in the network, and routing is performed in the direction of this vector field at every location of the network. The magnitude of the vector field at every location represents the density of amount of data that is being transited through that location. We define the total communication cost in the network as the integral of a quadratic form of the vector field over the network area. Our mathematical machinery is based on partial differential equations analogous to the Maxwell´s equations in electrostatic theory. We use our vector field model to solve the optimization problem for the case in which there are multiple destinations (sinks) in the network. In order to optimally determine the destination for each sensor, we partition the network into areas, each corresponding to one of the destinations. We define a vector field, which is conservative, and hence it can be written as the gradient of a scalar function (also known as a potential function). Then we show that in the optimal assignment of the communication load of the network to the destinations, the value of that potential function should be equal at the locations of all the destinations. Also, we show that such an optimal partitioning of the network load among the destination is unique, and we give iterations to find the optimal solution