On communication over gaussian sensor networks with adversaries: Further results

This paper presents new results on the game theoretical analysis of optimal communications strategies over a sensor network model. Our model involves one single Gaussian source observed by many sensors, subject to additive independent Gaussian observation noise. Sensors communicate with the receiver over an additive Gaussian multiple access channel. The aim of the receiver is to reconstruct the underlying source with minimum mean squared error. The scenario of interest here is one where some of the sensors act as adversary (jammer): they aim to maximize distortion. While our recent prior work solved the case where either all or none of the sensors coordinate (use randomized strategies), the focus of this work is the setting where only a subset of the transmitter and/or jammer sensors can coordinate. We show that the solution crucially depends on the ratio of the number of transmitter sensors that can coordinate to the ones that cannot. If this ratio is larger than a fixed threshold determined by the network settings (transmit and jamming power, channel noise and sensor observation noise), then the problem is a zero-sum game and admits a saddle point solution where transmitters with coordination capabilities use randomized linear encoding while the rest of the transmitter sensors is not used at all. Adversarial sensors that can coordinate generate identical Gaussian noise while other adversaries generate independent Gaussian noise. Otherwise (if that ratio is smaller than the threshold), the problem becomes a Stackelberg game where the leader (all transmitter sensors) uses fixed (non-randomized) linear encoding while the follower (all adversarial sensors) uses fixed linear encoding with the opposite sign.