Optimal Node Density for Detection in Energy-Constrained Random Networks

The problem of optimal node density maximizing the Neyman-Pearson detection error exponent subject to a constraint on average (per node) energy consumption is analyzed. The spatial correlation among the sensor measurements is incorporated through a Gauss-Markov random field (GMRF) model with Euclidean nearest-neighbor dependency graph. A constant density deployment of sensors under the uniform or Poisson distribution is assumed. It is shown that the optimal node density crucially depends on the ratio between the measurement variances under the two hypotheses and displays a threshold behavior. Below the threshold value of the variance ratio, the optimal node density tends to infinity under any feasible average energy constraint. On the other hand, when the variance ratio is above the threshold, the optimal node density is the minimum value at which it is feasible to process and deliver the likelihood ratio (sufficient statistic) of the sensor measurements to the fusion center. In this regime of the variance ratio, an upper bound on the optimal node density based on a proposed 2-approximation fusion scheme and a lower bound based on the minimum spanning tree are established. Under an alternative formulation where the energy consumption per unit area is constrained, the optimal node density is shown to be strictly finite for all values of the variance ratio and bounds on this optimal node density are provided.