Filling Gap between Discrete and Continuous Space Flow Models in Dense Wireless Networks

Abstract—Optimizing information flow in a dense wireless network using discrete methods can be computationally pro-hibitive. Instead of treating the nodes as discrete entities, these networks can be modeled as continuum of nodes providing a medium for information transport. To model information routes in continuous space, information flow vector field is defined over the geographical domain of the network. At each point of the network, the orientation of the vector field shows the direction of the flow of information, and its magnitude shows the density of information flow. Using multivariate calculus techniques in continuous domain, an information flow vector field can be found such that it minimizes a suitable cost function. Then the solution is discretized. Conventionally, a centralized method of calculating the optimal information flow in the network is suggested; however, using a centralized method to optimize in-formation flow in a dynamic network is prohibitive. Additionally, the value of information flow vector field is needed only at the locations of nodes in the network. This poses a gap between the continuous space and discrete space models of information flow in dense wireless networks. This gap is how to calculate and apply the optimum information flow derived in continuous domain in a network with finite number of nodes. As a first step to fill this gap, a specific quadratic cost function is considered. It is proved that the the vector field that minimizes this cost function is irrotational, thus it is written as the gradient of a potential function. This potential function satisfies a Poisson Partial Differential Equation (PDE) which in conjunction with Neumann boundary condition has a unique solution up to a constant. The PDE resulted by optimization in continuous domain is first discretized and then solved in a distributed fashion. The solution requires only neighboring nodes to communicate with each other. The gradient of the resulting potential defines the routes that the traffic should be forwarded. I