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Network invariants for optimal input detection

This paper studies a detection problem for network systems, where changes in the statistical properties of an input driving certain network nodes has to be detected by sparse and remotely located sensors. We explicitly derive the Maximum A Posteriori (MAP) detector, and characterize its performance as a function of the network parameters, and the location of the sensor nodes. We show that, in the absence of measurement noise, the detection performance obtained when sensors are located on a network cut is not worse than the performance obtained by measuring all nodes of the subnetwork induced by the cut and not containing the input node. Conversely, in the presence of measurement noise, we show that the detection performance may increase or decrease with the graphical distance between the input node and the sensors. We view the propagative properties of the network as an invariant enforced by the structure and weights, and we remark that such invariant properties may be effectively used for the design and operation of secure cyber-physical systems.